Custom Theme Automatic Updates

My company runs about 20 internal sites that all use the same template (some have different child themes). We are constantly updating the theme to the tune of a new release about every 2 weeks.

Right now as you can imagine it’s a bit of work uploading the new theme folder to 20 sites. I was hoping that I would be able to setup the theme to automatically update when we post a new version (just like the twentyfifteen theme) but I can’t seem to find any information on it.

Can someone point me in the right direction? Thanks.