There is an interesting paper of Adem:

A. Adem: Automorphisms and Cohomology of Discrete Groups, J. of Algebra 182(1996), 721-737,

where he considers non-abelian cohomology with coefficients in a discrete group $\Gamma$ of finite cohomological dimension (in particular $\Gamma$ is torsion-free).

Among others he cites (and gives a proof) of the following result due to Serre:

1) Let $G$ be a finite subgroup of $\operatorname{Aut}(\Gamma)$, let $\bar{\Gamma} = \Gamma \rtimes G$ be the semi-direct produt and let $\kappa: \bar{\Gamma} \to G$ be the natural projection. Then for each $H \le G$ there is a bijection between the conjugacy classes $\lbrace \bar{H} \le \bar{\Gamma} | \kappa(\bar{H}) = H \rbrace / \Gamma$ and $H^1(H;\Gamma)$.

In particular $H^1(G;\Gamma)$ is finite iff there are (up to $\Gamma$-conjugacy) only finitely many $\bar{G} \le \bar{\Gamma}$ such that $\bar{G}\Gamma/\Gamma \cong G$.

Adem uses this result to prove:

2) If $P$ is a $p$-subgroup of $\operatorname{Aut}(\Gamma)$ then $|H^1(P;\Gamma)| \le \dim_{\mathbb{F}_p} H^{\ast}(\Gamma;\mathbb{F}_p)$. In particular $H^1(P;\Gamma)$ is finite.

Perhaps you can adjust some of the methods used there to the situation you have in mind.