Release, debug version and Google Authorization?

Question

Google

The bottom line is that there are official Google sources about authorization through it ... The problem is that if you run the debug version, it works regardless of the device (which is excellent on the emulator, as on a real phone, tested on 4.1 and 5.1). But the main trouble with the release version, it already works more problems only on some phones version < 4.1.
As one woodsy toaster user already answered:

The release version does not work, because it is signed by another certificate, and its fingerprint is most likely not registered in your Developer Console

Please tell me how to solve this problem...