The famous Harer stability theorem asserts that the homology group $H_d(\mathcal{M}_{g,n},\mathbf{Z})$ is independent of *g* and *n* in the range $0 \leq 2d < g-1$. This is proven by analyzing the maps of mapping class groups $\Gamma_{g,n}\to \Gamma_{g+1,n}$ given by gluing a torus with a disk removed to a boundary circle (when $n \geq 1$), and $\Gamma_{g,n} \to \Gamma_{g,n-1}$ by gluing in a disk, and showing that these maps induce an isomorphism on homology in low dimensions (regardless of the choices involved in writing down such maps).

By considering curves with level structures, one obtains finite covers of $\mathcal{M}_{g,n}$, or equivalently, finite index subgroups of the mapping class group. So let's consider a finite group *G*, and denote by $\mathcal{M}_{g,n}[G]$ the moduli space parametrizing *n*-pointed smooth curves of genus *g* equipped with an étale *G*-torsor. Is the corresponding statement for $H_d(\mathcal{M}_{g,n}[G],\mathbf{Z})$ known? It is not hard to write down analogues in this context of the maps of mapping class groups above.

Remark: The corresponding statement for moduli of *spin curves* is known (and is also a theorem of Harer), so one might hope for a statement like this because of the similarities between the spaces of *r*-spin curves and the spaces of curves with $G=\mathbf{Z}/r\mathbf{Z}$ level structure.