  In an earlier paper the notion of a filtered derived equivalence was
introduced, and it was shown that if two K3 surfaces admit such an equivalence
then they are isomorphic. In this paper we study more refined aspects of
filtered derived equivalences related to the action on the cohomological
realizations of the Mukai motive. It is shown that if a filtered derived
equivalence between K3 surfaces also preserves ample cones then one can find an
isomorphism that induces the same map as the equivalence on the cohomological
realizations.
