Originally Posted by
Ryllharu
The only thing V.V. told Suzaku was that Zero had a power called the Geass [Episode 24-25 starting at 14:22]. Not who Zero was, and gave him no instructions. Just like C.C. doesn't tell Lelouch what to do, though we really have no idea what she wants in exchange yet or when she will demand payment. But V.V. doesn't have a deal with Suzaku as Lelouch does with C.C. There is no obligation between them. He can't give Suzaku any orders. V.V. can manipulate him, and he surely did that. In no way, shape, or form did V.V. tell Suzaku to do anything. Suzaku went to take revenge for Euphie. No one else.
There's no reason to believe that he wouldn't now that he takes orders only from the royalty, maybe only the Emperor himself. The Emperor showed clear motive for wiping out the entirety of the Black Knights, and clearly wanted C.C. dead. Why give her any additional aid allowed to operate freely? As half-Britannian, particularly higher society, Kallen would still have free access if she wasn't outed as a member of the Black Knights.