Adding AD Security group to a SQL linked server
With groups it should be work in the same way as single user.
You must have in place proper spn to grant kerberos delegation between the two  instances.
Maybe you have tried from ssms so you haven't done the two hops for authentication. In that way NTLM is granting you permission but when you are using an application you will need to grant authentication on two hops and so you fall on the anonymous authentication error that is not granting your user proper login.
Please check it.
https://docs.microsoft.com/en-us/sql/database-engine/configure-windows/register-a-service-principal-name-for-kerberos-connections?view=sql-server-ver15