There exist a variety of ways to do this. How complicated do you want it to be, and how many other concerns does this implicate? Does Gamma have to auth as actually being Gamma to Beta or is that out of scope? etc

The most straightforward way to be is (duh duh) public key crypto. GPG is fine. Gamma talks to Alpha. Alpha tells Gamma: "Give this message to Beta: 'I trust Gamma. Nonce: whatever.' Here's the corresponding signature." Beta has pre-arranged to have Alpha's public key. On presentation of Alpha's message from Gamma and verification of the signature, Beta extends trust to Gamma.

There are a variety of ways to screw this up. Don't roll your own crypto primitives. It is highly likely you have consequential requirements which break the design of this cryptosystem, like e.g. requiring the ability for Alpha to say "Nah, actually, changed my mind about Gamma. They're not authorized anymore."