The security group in AD becomes a LOGIN at the SQL Server level. Adding a new database won't do anything to create a USER associated with that LOGIN. What I can suggest, as a quick fix, is to script out a USER from an existing database and then apply that script whenever you create a new database. The restore of the database is a slightly different story. I'm assuming that the AD group is already a login on your test server. It could be that the IDs of the LOGIN and database USER are out of sync, even though the names are in agreement. If so, you could re-connect the two by running:
ALTER USER MyUser WITH LOGIN = MySecurityGroup;Again, you could script out this code for the full set of AD groups you are using and run it against any restored database.
There are two kinds of light -- the glow that illuminates, and the glare that obscures. -James Thurber