Git doesn't keep an audit trail of the operations done to a repo.
Plus it doesn't have any authentication mechanism, so it doesn't know who did what.
To keep track of that (who did what), you need to add to your central a repo an authorization layer like gitolite.
That assumes you communicate with your central repo through an authentication mechanism like:
- Apache
- SSH
(using a shared path or git url wouldn't include the authentication id that gitolite needs to validate/authorize the git command)
The idea is that, since gitolite validate every git commands based on who you are, it is in a unique position to keep the kind of audit trail (who did what) you are after.