How do you compute group cohomology in practice?

If you have a finite group $G$ and a finite $G$-module $K$, and you need to know $H^1(G,K)$ or $H^2(G,K)$, how do you do it? Do you use a computer algebra system? (If so, which one?) Do you use a short exact sequence $0\rightarrow A\rightarrow K\rightarrow B\rightarrow 0$ of $G$-modules for which you can find $H^*(G,A),H^*(G,B)$ more easily? Do you use some sort of resolution? Something else?

Motivation: I am studying small metabelian $2$-groups and it would be very handy, given two abelian groups $Q,K$ with a $Q$-module structure on $K$, to be able to know $H^2(Q,K)$. I did this by hand, directly writing down the 2-cocycles, for $Q=\mathbb{Z}/2, K=\mathbb{Z}/4$, but already with $Q,K$ both the Klein 4-group, I am finding it impractical.

Solution 1:

There is a very usefull program for discrete algebra called GAP. I am not sure if it already has a method for what you need. But it is free and it is very useful for many computations, so I recommend you try it out.

Solution 2:

Magma is becoming my go-to for this.