Big monodromy for higher Prym representations

Let $\Sigma_{g'}\to \Sigma_g$ be a cover of an orientable surface of genus g by an orientable surface of genus g', branched at n points, with Galois group H. Such a cover induces a virtual action of the mapping class group $\text{Mod}_{g,n+1}$ of a genus g surface with n+1 marked points on $H^1(\Sigma_{g'}, \mathbb{C})$. When g is large in terms of the group H, we calculate precisely the connected monodromy group of this action. The methods are Hodge-theoretic and rely on a "generic Torelli theorem with coefficients."