  Fix a finite group $G$ and a conjugacy invariant subset $C\subseteq G$. Let
$\Sigma$ be an oriented surface, possibly with punctures. We consider the
question of when two homomorphisms $\pi_1(\Sigma) \to G$ taking punctures into
$C$ are equivalent up to an orientation preserving diffeomorphism of $\Sigma$.
We provide an answer to this question in a stable range, meaning that $\Sigma$
has enough genus and enough punctures of every conjugacy type in $C$. If $C$
generates $G$, then we can assume $\Sigma$ has genus 0 (or any other constant).
The main tool is a classifying space for (framed) $C$-branched $G$-covers, and
related homology classes we call branched Schur invariants, since they take
values in a torsor over a quotient of the Schur multiplier $H_2(G)$. We
conclude with a brief discussion of applications to $(2+1)$-dimensional
$G$-equivariant TQFT and symmetry-enriched topological phases.
