automatic pushing from github to a remote url

Question

I'm using git to manage my WordPress theme my hosting (WPEngine) provides a git push option that allows me to publish my git repo to the server.

I want to manage my repo in GitHub and I was wondering if there is a way to use it like the Heroku integration with GitHub (automatic deployment whenever changes are made in a specific branch)

ideally, what I need is to publish the master branch to a remote whenever its header is changed (whenever changes are made) the alternative is to push to multiple remotes (both GitHub and wpengine) whenever I make changes and this will be messy and I will lose the GitHub code managing and pull requests etc...)