${\operatorname{Mod}}(S)$ in the center $Z(G)$ of $G$ and $G\cdot{\operatorname{Mod}}(S)^G$ is a subgroup of the normalizer of $G$ in ${\operatorname{Mod}}(S)$. The center $Z(G)$ acts trivially on ${\operatorname{Teich}}(S)^G$ and the quotient ${\operatorname{Mod}}(S)^G/Z(G)$ can be identified with a finite index subgroup of the mapping class group ${\operatorname{Mod}}(S_G)$ (this is the group of mapping classes of the underlying surface which respect the type of the orbifold points). Note that an element of ${\operatorname{Mod}}(S_G)$ lies in this image if and only if it lifts to a mapping class of $S$ which commutes with the $G$-action.

The group $G\cdot{\operatorname{Mod}}(S)^G\subset {\operatorname{Mod}}(S)$ acts on the restriction ${{\mathscr C}}_{{\operatorname{Teich}}(S)^G}/{\operatorname{Teich}}(S)^G$ of ${{\mathscr C}}(S)/{\operatorname{Teich}}(S)$ to ${\operatorname{Teich}}(S)^G$ and since $G$ acts trivially on ${\operatorname{Teich}}(S)^G$, this turns the restriction ${{\mathscr C}}_{{\operatorname{Teich}}(S)^G}/{\operatorname{Teich}}(S)^G$ into a $G$-curve over ${\operatorname{Teich}}(S)^G$. It is universal as a marked $G$-curve, the marking being given by an equivariant isotopy class of diffeomorphisms with $(S,G)$ (so $\phi$ is here still being suppressed in the notation). If we divide out this universal marked $G$-curve by the action of ${\operatorname{Mod}}(S)^G$, that is, if we pass to the stack or orbifold quotient, then we obtain a smooth complex D-M stack, denoted ${{\mathscr M}}^\phi$, which parameterizes the $G$-curves of topological type $\phi$. It is characterized by 