The group wiki describes the cohomology groups $H^n(G,M)$ where $G=(\mathbb{Z}/p\mathbb{Z})^2$, $p$ is prime (I'm not sure how much this matters for that computation, but let's assume it), and $M$ has trivial $G$-action.
Specifically, if $n>0$: $$H^n(G,M) \approx \left\{ \begin{array}{lc} M[p]^{\frac{n+3}{2}}\oplus (M/pM)^{\frac{n-1}{2}} & \text{if $n$ is odd} \\ M[p]^{\frac{n}{2}}\oplus (M/pM)^{\frac{n+2}{2}} & \text{if $n$ is even} \end{array} \right.$$ where $M[p]$ is the $p$-torsion subgroup of $M$.
This involves various manipulations with the Künneth formula and the dual universal coefficient theorem, and the combinatorics become a little messy, especially since it requires some non-natural splitting of short exact sequence.
Now any $g\in G$ defines a morphism $f_g: \mathbb{Z}/p\mathbb{Z}\to G$, and thus by restriction, a morphism $$(f_g)^*: H^n(G,M)\to H^n(\mathbb{Z}/p\mathbb{Z}, M).$$ The good thing is that on the other hand $H^n(\mathbb{Z}/p\mathbb{Z}, M)$ is easy to understand: it is $M[p]$ when $n$ is odd, and $M/pM$ when $n$ is even (and nonzero).
Is there a natural way to describe the maps $$ M[p]^{\frac{n+3}{2}}\oplus (M/pM)^{\frac{n-1}{2}} \to M[p] $$ (for $n$ odd) and $$ M[p]^{\frac{n}{2}}\oplus (M/pM)^{\frac{n+2}{2}} \to M/pM$$ (for $n$ even) induced by each $g\in G$ (even though there are some non-natural choices in the description of the groups)?
As a corollary of how well they can be described, do they depend "nicely" on $g$? (Like, additively?)
For instance, in the case $n=1$, we are looking at a map $M[p]^2=\operatorname{Hom}(G,M)\to M[p]$, and this is just the evaluation at $g$.
I should specify that I am mainly interested in the case $n=3$, but I think understanding the general combinatorics can shed some light on the low-degree cases.