Database Migration - Create New Filegroup and Move Objects
This can be done if you have clustered indexes (either as the primary key constraint or on their own) on the tables that you wish to move.

So restore the database, add the new filegroup and then...

For primary keys:-

 ALTER TABLE [SCHEMA].[TABLE NAME] DROP CONSTRAINT [CONSTRAINT NAME]

 ALTER TABLE [SCHEMA].[TABLE NAME] ADD PRIMARY KEY CLUSTERED 
 (
   [Column]  ASC
 )WITH (PAD_INDEX = OFF, STATISTICS_NORECOMPUTE = OFF, SORT_IN_TEMPDB = OFF, IGNORE_DUP_KEY = OFF, 
    ALLOW_ROW_LOCKS = ON, ALLOW_PAGE_LOCKS = ON) ON [NEW FILEGROUP]
 GO


For clustered indexes:-

CREATE CLUSTERED INDEX [CONSTRAINT NAME] ON [SCHEMA].[TABLE NAME]
(
    [Column] ASC
)WITH (PAD_INDEX = OFF, STATISTICS_NORECOMPUTE = OFF, SORT_IN_TEMPDB = OFF, IGNORE_DUP_KEY = OFF, DROP_EXISTING = ON, 
    ALLOW_ROW_LOCKS = ON, ALLOW_PAGE_LOCKS = ON) ON [NEW FILEGROUP]
GO


Please be warned that this may take a while to do on a VLDB!