ShiftEdit can connect to a git repository, show the history and create new commits.
In order for this to work you need to use SFTP or FTP (via turbo mode) and the .git folder must reside in the root.
SFTP works straight-forwardly by executing Git commands over SSH.
If using the FTP proxy, Git commands will be executed by the PHP proxy script. For this to work the files will need to be writable by the web user (e.g. group or 777 permissions). If the files are not writable then the script will attempt "sudo" with the file owner name.
Sudo access is preferable as it is more secure. To enable sudo you will need to add a rule to sudoers by doing something like this:
# nano /etc/sudoers.d/shiftedit
apache ALL=(myusername) NOPASSWD: /usr/bin/git
Replace "apache" with your web user if different and replace "myusername" with a user that has write permissions to the files. Save with Ctrl-o, then Ctrl-x to exit.