Let $G$ be a finite group, and let $0\to M_1\xrightarrow{\iota} M_2\xrightarrow{\pi} M_3\to 0$ be a short exact sequence of $G$-modules (finitely generated over $\mathbb Z$, not necessarily free). I have a description of my group in terms of generators and relations ($G$ is quite small), I have matrices describing the action of $G$ on the $M_i$, and matrices describing the homomorphisms $\iota$ and $\pi$ with respect to some bases of $M_i$. I would like to understand the surjectivity of the map $H^1(G,M_2)\to H^1(G,M_3)$, or equivalently whether $H^1(G,M_3)\to H^2(G,M_1)$ is the zero map or not.

I am interested in this calculation for many different sequences, some of which are not so nice, so I don't expect to be able to do this by hand (e.g. using some trick or nice observation).

Is there some way to do this with a computer?