the genus of $S$) comes with an action of ${\operatorname{Aut}}(G)$, which permutes its irreducible components through ${\operatorname{Out}}(G)$. If we pass to the stack quotient with respect to this ${\operatorname{Aut}}(G)$-action, we obtain a D-M stack ${{\mathscr M}}^{(G)}$ which parameterizes curves $C$ endowed with a subgroup of ${\operatorname{Aut}}(C)$ isomorphic to $G$ (but with no such isomorphism specified).

The forgetful morphism ${{\mathscr M}}^{\phi}\to {{\mathscr M}}(S)$ is finite and the universal $G$-curve ${{\mathscr C}}_{{{\mathscr M}}^\phi}/{{\mathscr M}}^\phi$ fits in a forgetful cartesian diagram which maps to the universal curve ${{\mathscr C}}_{{{\mathscr M}}(S)}/{{\mathscr M}}(S)$. Note that we have a factorization ${{\mathscr M}}^{\phi}\to{{\mathscr M}}^{(G)}\to {{\mathscr M}}(S)$. In fact, ${{\mathscr M}}^{(G)}$ can be regarded as the normalization of the image of ${{\mathscr M}}^{\phi}\to {{\mathscr M}}(S)$ (this expresses the fact that a curve $C$ whose automorphism group contains a copy of $G$, will in general have only one such copy) and ${{\mathscr M}}^{\phi}\to{{\mathscr M}}^{(G)}$ will be étale.

The following observation is important for what follows. For every $G$-curve $(C,G)$ of type $\phi$ (so defining a point of ${{\mathscr M}}^\phi$), $G$ acts on $H^0(C,\Omega_C)$ but the character of this action will be independent of the point we choose, simply because ${{\mathscr M}}^\phi$ is irreducible. In other words, this only depends 