  We compute cohomology of the moduli space of genus three curves with level
two structure and some related spaces. In particular, we determine the
cohomology groups of the moduli space of plane quartics with level two
structure as representations of the symplectic group on a six dimensional
vector space over the field of two elements. We also make the analogous
computations for some related spaces such as moduli spaces of genus three
curves with a marked points and strata of the moduli space of Abelian
differentials of genus three.
