Let $\Gamma$ be a cocompact Fuchsian group. So it has presentation $$\langle x_1,y_1, \dots, x_g,y_g,z_1, \ldots, z_r \mid [x_1,y_1] \cdots [x_g,y_g]z_1 \cdots z_r=1, \ z_i^{m_i}=1 \rangle$$ for some $g \geq 2$, $r \geq 0$ and $m_i \geq 2$.

Question:Does there exist some $n \geq 1$ such that $\mathrm{Aut}(\Gamma)$ contains a finite-index subgroup isomorphic to $\mathrm{Mod}(S_{g,n})$?

The case $r=0$ can be found in Farb and Margalit's book *A primer on mapping class groups* (page 235). There, it is explained how to deduce from the Dehn-Nielsen-Baer theorem and the Birman exact sequence that $\mathrm{Aut}(\pi_1(S_g))$ is isomorphic to $\mathrm{Mod}^{\pm}(S_{g,1})$.

On the other hand, a Dehn-Nielsen-Baer theorem for Fuchsian groups can be found in MacLachlan and Harvey's article *On mapping class groups and Teichmüller spaces*, where it is proved that $\mathrm{Out}(\Gamma)$ is virtually isomorphic to some $\mathrm{Mod}(S_{g,m})$.

A natural guess is that $\mathrm{Aut}(\Gamma)$ should be virtually isomorphic to $\mathrm{Mod}(S_{g,m+1})$, with essentially the same arguments but replacing surfaces with 2-orbifolds.

It is probably pretty standard, but I would need a reference to cite in a paper I am writing.

Surfaces and Planar Discontinuous Groups, written by Heiner Zieschang, Elmar Vogt and Hans-Dieter Coldewey, I found an entire chapter dedicated to automorphism groups of planar groups. The result I am looking for does not seem to be written there, but it seems to be good starting point. $\endgroup$ – AGenevois Oct 26 '18 at 14:44