  This note is about the Chow groups of a certain family of smooth cubic
fourfolds. This family is characterized by the property that each cubic
fourfold $X$ in the family has an involution such that the induced involution
on the Fano variety $F$ of lines in $X$ is symplectic and has a $K3$ surface
$S$ in the fixed locus. The main result establishes a relation between $X$ and
$S$ on the level of Chow motives. As a consequence, we can prove
finite-dimensionality of the motive of certain members of the family.
