the property that if $Y$ is a complex variety (or scheme for that matter), then a smooth $G$-curve over $Y$ of topological type $\phi$ gives rise to a (stack) morphism $Y\to {{\mathscr M}}^\phi$, and that thus is obtained a bijection between the isomorphism classes of smooth $G$-curves over $Y$ of topological type $\phi$ and the morphisms from $Y$ to ${{\mathscr M}}^\phi$. Beware that if we ignore the stack structure by passing to the underlying variety, the fiber over every closed point is a quotient of a $G$-curve by a subgroup of $G$ which contains $Z(G)$.

It is clear that ${{\mathscr M}}^\phi$ only depends on the ${\operatorname{In}}({\operatorname{Mod}}(S))$-orbit of $\phi$. Since ${\operatorname{Teich}}(S)^G$ is a simply-connected manifold, ${{\mathscr M}}^\phi$ is irreducible and the fundamental group of the underlying analytic orbifold is ${\operatorname{Mod}}(S)^G$. Precomposition with an automorphism $\rho$ of $G$ may lead to a different D-M complex stack ${{\mathscr M}}^{\phi\rho}$ to which we shall refer as the *$\rho$-twist* of ${{\mathscr M}}^\phi$. If however, $\rho$ is inner and defined by conjugation with some $g\in G$, say, then $g: C\to C$ makes the $G$-curve isomorphic to its $\rho$-twist. So the union ${{\mathscr M}}^{[G]}$ of ${\operatorname{Aut}}(G)$-twists (denoted by Collas-Maugeais in [@cm] by ${{\mathscr M}}_{g}[G]$, where $g$ is 