Database mappings are actually relationships between the syslogins table in the master database and sysusers table in the user database. If there is no record for a Windows user or group in the database when you restore it, then the user or group does not get any permissions and appears as though there is no mapping.

Your thinking is correct - you either add the group to the database on serverB or run a script to add the user to the database after it has been restored to serverA.