Let $M=\mathbb{Z}/2\mathbb{Z}$.
(a) Show there is an isomorphism of abelian groups $H^2(M\oplus M,M)\cong M \oplus M\oplus M$, where $H^2$ denotes for the second cohomology group.
(b) Let $G=GL_2(\mathbb{F}_2)$ act on $M\oplus M$ as automorphisms in the natural way. Show that we have an action of $G$ on $Z^2(⊕,)$ given by $$(\alpha . f)(a_1,a_2)=f(a_1\alpha^{-1},a_2\alpha^{-1})$$ for each $\alpha \in G$, $a_i \in M\oplus M$ and $f\in Z^2(⊕,)$. Furthermore show this defines an action on $H^2(M\oplus M,M)$
(c) Calculate the orbits of G on $H^2(M\oplus M,M)$.
First of all I'm trying to explicitly find all 2-cocycles and 2-coboundries from $M\oplus M \times M\oplus M$ to $M$ and then find $Z^2$ and $H^2=Z^2/B^2$ but I don't see a straightforward way to do so.