Let $S$ be a closed oriented surface and $C(S, n)$ be the configuration space of $n$ points on $S$, i.e., the space of $n$-tuples of distinct points of $S$ with the topology induced from $S^n$. Let $V \subset S$ be a set of $n$ distinct points. Let ${\rm Homeo}_0(S)$ be the topological group (with compact-open topology) of homeomorphisms $S \rightarrow S$ isotopic to identity, and ${\rm Homeo}_0(S, V)$ be the subgroup of those that fix $V$ and can be isotopic to identity by an isotopy fixing $V$. Denote by $H(S, V)$ the topological space obtained in the quotient of ${\rm Homeo}_0(S)$ by ${\rm Homeo}_0(S, V)$.

Every homeomorphism $h: S \rightarrow S$ extends to a homeomorphism $h_n: C(S, n) \rightarrow C(S, n)$, and eventually to a homeomorphism $\tilde h_n: \widetilde{C(S, n)} \rightarrow \widetilde{C(S, n)}$ of the universal cover of $C(S, n)$. By fixing an arbitrary lift $\tilde V \in \widetilde{C(S, n)}$ of $V$ and noting that the extensions of homeomorphisms in ${\rm Homeo}_0(S, V)$ fix $\tilde V$ we get a continuous map from $H(S, V)$ to $\widetilde{C(S, n)}$. My intuition says that this is a homeomorphism (and I can confirm it for $n=1$). Does anyone know a reference for this?