  Let $X\subset \P^5$ 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 $\sA_X \simeq
D^b(S,\alpha)$ where $\sA_X$ is the Kuznetsov component of $D^b(X)$ and
$\alpha$ is a Brauer class, or if there is an isomorphism between the
transcendental motive $t(X)$ and the (twisted ) transcendental motive of a K3
surface$S$. 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.
