I believe what is happening is that you did not send the "become a
tester" link to your testers for them to click.
Apparently the part where you add the Google Group or Google+ Community
to the Alpha/Beta testers on the Google Play Developers console you did all
But also there on the console you'll get a link that is:
https://play.google.com/apps/testing/<your app package
all those users/testers MUST click this link and click on the button
"Become a tester" for it to work.
After that initial setup is done, your users should receive automatic
updates just like normal production version.
hope it helps.