# Generalizations of Dehn-Nielsen-Baer for topological branched cover?

For any manifolds $M$, a homotopy class of diffeomorphism gives rise to an automorphism of $\pi_1(M)$ (up to conjugacy since we are dealing with free homotopies). Moreover, in the specific case of surfaces, Dehn-Nielsen-Baer's theorem tells us that the map $\operatorname{Mod}(M) \to \operatorname{Out}(\pi_1(M))$ is an isomorphism. This is actually really specific to surfaces and do not hold in higher dimensions in general. However, I am wondering if this theorem could be generalized to topological branched cover of a given degree.

This indeed works in the case of a branched cover of the sphere as the homotopy type of the map is determined by its degree.

So more generally, if $f,g : \Sigma_h \to \Sigma_g$ are a topological branched cover of surfaces such that the induced maps on the fundamental groups are the same up to conjugacy, then is $f \simeq g$? If so, is there a good reference where I could read about it? If not, is there an easy counter example?

I was thinking that a possible counter example might come from precomposing $f$ with a Dehn twist $m$ around an essential curve $\gamma$ such that $f(\gamma) \simeq \ast,$ but I haven't work out this example in detail yet.

Edit: It may be worth mentioning that the homotopies that I am considering are not relative to the branched points. In particular, the branching data is not necessarily fixed by the homotopy class.

This is true in general, since $\Sigma_g$ is a $K(\pi,1)$ for $g\ge 1$, and if $A$ and $B$ are groups then the set of unbased homotopy classes $[K(A,1),K(B,1)]$ is in one-to-one correspondence with homomorphisms from $A$ to $B$ modulo conjugacy in $B$. This latter fact is well-known, and follows from Proposition 4A.2 of Hatcher's textbook.