Updating GExtension is easy. Just navigate to Admin -> Settings -> Update.
For your own safety: Check the changelog (by clicking on the available version) before updating. If there is no changelog, wait with updating until it's out. The changelog sometimes brings important notices and verifies that the update is official.
Press the "Update" button if an update is available and wait until the process is finished.
Important: Sometimes is it required to clear the browser cache for all pages to work correctly. Press CTRL+F5 on pages that do not display correctly. You may also need to clear the cache on services like CloudFlare.
Press the "Lua" button and save the ZIP file on your computer. Unzip the files and delete the gextension/lua/gextension/config folder (you may want to keep this folder and reconfigure GExtension if there are any new options). New options will be mentioned in the changelog and it's always save to delete the config folder for updating. Connect to your GMOD server file system and delete all subfolders of addons/gextension except gextension/lua/gextension/config. Now, upload the new files to the addon directory.
There may will be an automatic FTP update in the future.