Google Play Alpha/Beta testing link is broken

Question

I have developed my app and have published it through Google Play for alpha testing. As one of the testers I get an opt-in link, where I signed in as a tester. After that I was hoping to download the app directly with my phone by going to the Play Store on my phone. But as it seems to turn out, I have to got to the app's web profile and then from there click download; only then does the Play Store download the app to my phone.

I want to get other alpha testers involved and I was hoping to make the process easier for them. Is there an easier way? As in, once they opt-in, I would like them to just use their Android phone to download the app the way they would any other app. Is there a way?

Clarifications:

• I am not looking for an alternative to Google Play

• I have already gone through the Google Group creation process. My question is concerning what happens after a group member opts-in. Can they use their phone directly (play-app on phone) or must they download from the website? So far I have not been successful using the play-app.