vmassuchetto

This is a plugin for managing website releases under production or development 
environments. Once the WordPress installation is in a Git tree, WP Deploy will 
detect its branches and put a switch menu in the admin bar.
Like this, you can just push your website to your remote and use the menu to 
deploy the changes without using Git hooks or prompting into another shell.
Another useful purpose is for the development process, where different 
implementations can be viewed and switched by customers, or a production branch 
can be testes in a staging environment.

WP Git Deploy

Deploy websites using Git and the admin bar.

PiotrPress

Clearcode

This plugin allows you to automatically send changes to your GIT repository, immediately after any update is made on your site.
If you store the source code of your website in the GIT repository, you can use this plugin to automatically send changes in the website’s code to the repository after each update. 
The plugin checks for updates (core, plugin, theme, translation) in a selected time interval, creates a new commit, and makes a push to the GIT repository. 
The plugin supports automatic WordPress updates. After each action the plugin saves the status to logs which are available through wp-admin and also are sent by email to defined recipients.

CC-Update

This plugin allows you to automatically send changes to your GIT repository, immediately after any update is made on your site.

WP Git Deploy vs CC-Update