This code should have the session variable name and the corresponding column name.
$Authenticate->storeResult("MM_Username", "MM_Username");
$Authenticate->storeResult("password", "password");
$Authenticate->storeResult("MM_UserGroup", "MM_UserGroup");
so it should be something like:
$Authenticate->storeResult("user_name", "MM_Username");
$Authenticate->storeResult("user_group", "MM_UserGroup");
I'm guessing your user group column name, but the point is that the column name is first and then the session variable name it is stored as.