Relating Hodge-theoretic K-theory of invariant categories to that of the original category

Develop a general, explicit formula describing the variation of Hodge structures Ktop[0](C^G/S), for a finite group G acting on a smooth proper S-linear category C of geometric origin, in terms of Ktop[0](C/S).

Background

The authors use the Hodge theory of smooth proper S-linear categories of geometric origin, encoded by the local system Ktop0. For invariant categories under finite group actions, understanding Ktop0 is crucial for applications, including equivariant semiregularity.

While they obtain rational identifications on invariant parts in special cases (e.g., for abelian groups and under suitable assumptions), a general simple relationship between Ktop0 and Ktop0 is not currently available.