K3 surfaces associated to a cubic fourfold

Let be a smooth cubic fourfold. A well known conjecture asserts that is rational if and only if there a Hodge theoretically associated K3 surface . The surface can be associated to in two other different ways. If there is an equivalence of categories where is the Kuznetsov component of and is a Brauer class, or if there is an isomorphism between the transcendental motive and the (twisted ) transcendental motive of a K3 surface . In this note we consider families of cubic fourfolds with a finite group of automorphisms and describe the cases where there is an associated K3 surface in one of the above senses.