Equivariant cohomology of the moduli space of genus three curves with symplectic level two structure via point counts

We determine the cohomology groups of the quartic and hyperelliptic loci inside the moduli space of genus three curves with symplectic level two structure as representations of the symmetric group S7 together with their mixed Hodge structures by means of making equivariant point counts over finite fields and via purity arguments. This determines the weighted Euler characteristic of the whole moduli space of genus three curves with level two structure.