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McDuff and Segal proved that unordered configuration spaces of open manifoldssatisfy homological stability: there is a stabilization map $\sigma: C_n(M)\toC_{n+1}(M)$ which is an isomorphism on $H_d(-;\mathbb{Z})$ for $n\gg d$. For afinite group $G$ and an open $G$-manifold $M$, under some hypotheses we definea family of equivariant stabilization maps $\sigma_{G/H}:C_n(M)\toC_{n+|G/H|}(M)$ for $H\leq G$. In general, these do not induce stability forBredon homology, the equivariant analogue of singular homology. Instead, weshow that each $\sigma_{G/H}$ induces isomorphisms on the ordinary homology ofthe fixed points of $C_n(M)$, and if the group is Dedekind (e.g. abelian), weobtain the following Bredon homological stability statement:$H^G_d(\bigsqcup_{n\geq 0}C_n(M))$ is finitely generated over$\mathbb{Z}[\sigma_{G/H} : H\leq G]$. This reduces to the classical statementwhen $G=e$.

Bredon homological stability for configuration spaces of $G$-manifolds