K3 surfaces associated to a cubic fourfold

Let X ⁵ be a smooth cubic fourfold. A well known conjecture asserts that X is rational if and only if there an Hodge theoretically associated K3 surface S. The surface S can be associated to X in two other different ways. If there is an equivalence of categories X Dᵇ (S, ) where X is the Kuznetsov component of Dᵇ (X) and is a Brauer class, or if there is an isomorphism between the transcendental motive t (X) and the (twisted) transcendental motive of a K3 surfaceS. 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.