(The question has been edited. It was pointed out in the comments that $\Gamma_G$ could be a surface group, thought of as a finite extension of another surface group, in which case $G$ is finite.)
Let $\pi$ be the fundamental group of a closed orientable surface, and let $G$ be a subgroup of $\mathrm{Out}(\pi)$.
Associated to $G$ is an extension
\begin{equation}
1 \to \pi \to \Gamma_G \to G \to 1
\end{equation}
which sits inside the sequence
\begin{equation}
1 \to \mathrm{Inn}(\pi) \to \mathrm{Aut}(\pi) \to \mathrm{Out}(\pi) \to 1.
\end{equation}
If there is a finite $K(G,1)$, it is not difficult to see that there is a finite $K(\Gamma_G,1)$. A colleague and I are interested in a converse.
If $G$ is torsion-free and there is a finite $K(\Gamma_G, 1)$, is there a finite $K(G,1)$?
I am aware of Wall's theorem that a finitely presented group $\mathcal{G}$ has a finite $K(\mathcal{G},1)$ if and only if it is of type (FL), meaning that there is a finite resolution of $\mathbb{Z}$ by finitely generated free $\mathcal{G}$-modules. A group $\mathcal{G}$ is of type (FP) if there is a finite resolution of $\mathbb{Z}$ by finitely generated projective $\mathcal{G}$-modules. As far as I know, there is still no known group of type (FP) that is not of type (FL).
I imagine that careful examination of the Hochschild-Serre spectral sequence may tell you that $G$ is of type (FP), but I didn't pursue this as I'd like to get all the way to (FL).
Is there some geometric construction I'm missing?
