Permissions on the git executable are fine; there is no reason why you will need to change this or change your group.
Most likely the permissions on your Git checkout are incorrect. If you accidentally checked this with something like sudo git clone ssh:// user@host /project.git or at some point you performed some operation in the repository, due to which some files will be owned by root, git works under your account, will not be able to write to these files.
Most likely, you need to make chown -R <user> myproject so that your project belongs to your own user account, not root.
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