  McDuff and Segal proved that unordered configuration spaces of open manifolds
satisfy homological stability: there is a stabilization map $\sigma: C_n(M)\to
C_{n+1}(M)$ which is an isomorphism on $H_d(-;\mathbb{Z})$ for $n\gg d$. For a
finite group $G$ and an open $G$-manifold $M$, under some hypotheses we define
a family of equivariant stabilization maps $\sigma_{G/H}:C_n(M)\to
C_{n+|G/H|}(M)$ for $H\leq G$. In general, these do not induce stability for
Bredon homology, the equivariant analogue of singular homology. Instead, we
show that each $\sigma_{G/H}$ induces isomorphisms on the ordinary homology of
the fixed points of $C_n(M)$, and if the group is Dedekind (e.g. abelian), we
obtain 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 statement
when $G=e$.
