Equivariant Kuznetsov components for cubic fourfolds with a symplectic involution

We study the equivariant Kuznetsov component ${\mathrm{K}\mathrm{u}}_G(X)$ of a general cubic fourfold $X$ with a symplectic involution. We show that ${\mathrm{K}\mathrm{u}}_G(X)$ is equivalent to the derived category $D^b(S)$ of a $K3$ surface $S$, where $S$ is given as a component of the fixed locus of the induced symplectic action on the Fano variety of lines on $X$.