WP_Automatic_Updater::send_email

			'auto_core_update_failed',
			array(
				'attempted'  => $core_update->current,
				'current'    => $wp_version,
				'error_code' => $error_code,
				'error_data' => $result->get_error_data(),
				'timestamp'  => time(),
				'retry'      => in_array( $error_code, $transient_failures, true ),
			)
		);

		if ( $send ) {
			$this->send_email( 'fail', $core_update, $result );
		}
	}

	/**
	 * Sends an email upon the completion or failure of a background core update.
	 *
	 * @since 3.7.0
	 *
	 * @param string $type        The type of email to send. Can be one of 'success', 'fail', 'manual', 'critical'.
	 * @param object $core_update The update offer that was attempted.
	 * @param mixed  $result      Optional. The result for the core update. Can be WP_Error.
	 */
	protected function send_email( $type, $core_update, $result = null ) {
		update_site_option(
			'auto_core_update_notified',
			array(
				'type'      => $type,
				'email'     => get_site_option( 'admin_email' ),
				'version'   => $core_update->current,
				'timestamp' => time(),
			)
		);

		$next_user_core_update = get_preferred_from_update_core();

		// If the update transient is empty, use the update we just performed.
		if ( ! $next_user_core_update ) {
			$next_user_core_update = $core_update;
		}

		$newer_version_available = ( 'upgrade' === $next_user_core_update->response && version_compare( $next_user_core_update->version, $core_update->version, '>' ) );

		/**
		 * Filters whether to send an email following an automatic background core update.
		 *
		 * @since 3.7.0
		 *
		 * @param bool   $send        Whether to send the email. Default true.
		 * @param string $type        The type of email to send. Can be one of
		 *                            'success', 'fail', 'critical'.
		 * @param object $core_update The update offer that was attempted.
		 * @param mixed  $result      The result for the core update. Can be WP_Error.
		 */
		if ( 'manual' !== $type && ! apply_filters( 'auto_core_update_send_email', true, $type, $core_update, $result ) ) {
			return;
		}

		switch ( $type ) {
			case 'success': // We updated.
				/* translators: Site updated notification email subject. 1: Site title, 2: WordPress version. */
				$subject = __( '[%1$s] Your site has updated to WordPress %2$s' );
				break;

			case 'fail':   // We tried to update but couldn't.
			case 'manual': // We can't update (and made no attempt).
				/* translators: Update available notification email subject. 1: Site title, 2: WordPress version. */
				$subject = __( '[%1$s] WordPress %2$s is available. Please update!' );
				break;

			case 'critical': // We tried to update, started to copy files, then things went wrong.
				/* translators: Site down notification email subject. 1: Site title. */
				$subject = __( '[%1$s] URGENT: Your site may be down due to a failed update' );
				break;

			default:
				return;
		}

		// If the auto-update is not to the latest version, say that the current version of WP is available instead.
		$version = 'success' === $type ? $core_update->current : $next_user_core_update->current;
		$subject = sprintf( $subject, wp_specialchars_decode( get_option( 'blogname' ), ENT_QUOTES ), $version );

		$body = '';

		switch ( $type ) {
			case 'success':
				$body .= sprintf(
					/* translators: 1: Home URL, 2: WordPress version. */
					__( 'Howdy! Your site at %1$s has been updated automatically to WordPress %2$s.' ),
					home_url(),
					$core_update->current
				);
				$body .= "\n\n";
				if ( ! $newer_version_available ) {
					$body .= __( 'No further action is needed on your part.' ) . ' ';
				}

				// Can only reference the About screen if their update was successful.
				list( $about_version ) = explode( '-', $core_update->current, 2 );
				/* translators: %s: WordPress version. */
				$body .= sprintf( __( 'For more on version %s, see the About WordPress screen:' ), $about_version );
				$body .= "\n" . admin_url( 'about.php' );

				if ( $newer_version_available ) {
					/* translators: %s: WordPress latest version. */
					$body .= "\n\n" . sprintf( __( 'WordPress %s is also now available.' ), $next_user_core_update->current ) . ' ';
					$body .= __( 'Updating is easy and only takes a few moments:' );
					$body .= "\n" . network_admin_url( 'update-core.php' );
				}

				break;

			case 'fail':
			case 'manual':
				$body .= sprintf(
					/* translators: 1: Home URL, 2: WordPress version. */
					__( 'Please update your site at %1$s to WordPress %2$s.' ),
					home_url(),
					$next_user_core_update->current
				);

				$body .= "\n\n";

				// Don't show this message if there is a newer version available.
				// Potential for confusion, and also not useful for them to know at this point.
				if ( 'fail' === $type && ! $newer_version_available ) {
					$body .= __( 'We tried but were unable to update your site automatically.' ) . ' ';
				}

				$body .= __( 'Updating is easy and only takes a few moments:' );
				$body .= "\n" . network_admin_url( 'update-core.php' );
				break;

			case 'critical':
				if ( $newer_version_available ) {
					$body .= sprintf(
						/* translators: 1: Home URL, 2: WordPress version. */
						__( 'Your site at %1$s experienced a critical failure while trying to update WordPress to version %2$s.' ),
						home_url(),
						$core_update->current
					);
				} else {
					$body .= sprintf(
						/* translators: 1: Home URL, 2: WordPress latest version. */
						__( 'Your site at %1$s experienced a critical failure while trying to update to the latest version of WordPress, %2$s.' ),
						home_url(),
						$core_update->current
					);
				}

				$body .= "\n\n" . __( "This means your site may be offline or broken. Don't panic; this can be fixed." );

				$body .= "\n\n" . __( "Please check out your site now. It's possible that everything is working. If it says you need to update, you should do so:" );
				$body .= "\n" . network_admin_url( 'update-core.php' );
				break;
		}

		$critical_support = 'critical' === $type && ! empty( $core_update->support_email );
		if ( $critical_support ) {
			// Support offer if available.
			$body .= "\n\n" . sprintf(
				/* translators: %s: Support email address. */
				__( 'The WordPress team is willing to help you. Forward this email to %s and the team will work with you to make sure your site is working.' ),
				$core_update->support_email
			);
		} else {
			// Add a note about the support forums.
			$body .= "\n\n" . __( 'If you experience any issues or need support, the volunteers in the WordPress.org support forums may be able to help.' );
			$body .= "\n" . __( 'https://wordpress.org/support/forums/' );
		}

		// Updates are important!
		if ( 'success' !== $type || $newer_version_available ) {
			$body .= "\n\n" . __( 'Keeping your site updated is important for security. It also makes the internet a safer place for you and your readers.' );
		}

		if ( $critical_support ) {
			$body .= ' ' . __( "If you reach out to us, we'll also ensure you'll never have this problem again." );
		}

		// If things are successful and we're now on the latest, mention plugins and themes if any are out of date.
		if ( 'success' === $type && ! $newer_version_available && ( get_plugin_updates() || get_theme_updates() ) ) {
			$body .= "\n\n" . __( 'You also have some plugins or themes with updates available. Update them now:' );
			$body .= "\n" . network_admin_url();
		}

		$body .= "\n\n" . __( 'The WordPress Team' ) . "\n";

		if ( 'critical' === $type && is_wp_error( $result ) ) {
			$body .= "\n***\n\n";
			/* translators: %s: WordPress version. */
			$body .= sprintf( __( 'Your site was running version %s.' ), get_bloginfo( 'version' ) );
			$body .= ' ' . __( 'We have some data that describes the error your site encountered.' );
			$body .= ' ' . __( 'Your hosting company, support forum volunteers, or a friendly developer may be able to use this information to help you:' );

			// If we had a rollback and we're still critical, then the rollback failed too.
			// Loop through all errors (the main WP_Error, the update result, the rollback result) for code, data, etc.
			if ( 'rollback_was_required' === $result->get_error_code() ) {
				$errors = array( $result, $result->get_error_data()->update, $result->get_error_data()->rollback );
			} else {
				$errors = array( $result );
			}

			foreach ( $errors as $error ) {
				if ( ! is_wp_error( $error ) ) {
					continue;
				}

				$error_code = $error->get_error_code();
				/* translators: %s: Error code. */
				$body .= "\n\n" . sprintf( __( 'Error code: %s' ), $error_code );

				if ( 'rollback_was_required' === $error_code ) {
					continue;
				}

				if ( $error->get_error_message() ) {
					$body .= "\n" . $error->get_error_message();
				}

				$error_data = $error->get_error_data();
				if ( $error_data ) {
					$body .= "\n" . implode( ', ', (array) $error_data );
				}
			}