Looking into this more I see that I can add an ADK to each group and it will automatically appear on the keyrings, but the problem with this is that the ADK added is just a public key and only good for encryption, not decryption.
I can add a key pair to each group as a group key, but I can't add the same key pair to every group and even if I could, the group key isn't automatically on the keychains.
What I need is for one user to encrypt data using their own key and a group keypair and have every other user in the organization be able to decrypt the file without using an ADK.