Update Installation Failed

This looks like a problem with file permissions.

Link to basic explanation: File Permissions Explained

I would contact your hosting company and ask them if you don’t know how to check and/or change this.

Please check Upgrading WordPress topic.

For Automatic Update to work, at least two criteria must be satisfied:

(a) file ownership: all of your WordPress files must be owned by the user under which your web server executes. In other words, the owner of your WordPress files must match the user under which your web server executes. The web server user (named “apache”, “web”, “www”, “nobody”, or some such) is not necessarily the owner of your WordPress files. Typically, WordPress files are owned by the ftp user which uploaded the original files. If there is no match between the owner of your WordPress files and the user under which your web server executes, you will receive a dialog box asking for “connection information”, and you will find that no matter what you enter in that dialog box, you won’t be able to update automatically.

(b) file permissions: all of your WordPress files must be either owner writable by, or group writable by, the user under which your Apache server executes.