I want to compute the group: $H^{2}(\mathbb{Z}_{2},\mathbb{Z}_{n})$ view in terms of normalized cochains, so as quotient of
$$ker\partial^{3}=\{f\mid f(x_{1},x_{2})-f(x_{1}+x_{2},x_{3})+f(x_{1},x_{2}+x_{3})-f(x_{1},x_{2})x_{3}=[0]_{n}\}$$
on
$$Im\partial^{2}=\{f\mid f(x_{1},x_{2})=g(x_{1})-g(x_{1}+x_{2})+g(x_{1})x_{2}\}$$ for a 1-cocycle $g$.
The main problem is how $\mathbb{Z}_{2}$ acts on $\mathbb{Z}_{n}$: we know that this means to fix a group homomomorphism $\mathbb{Z}_{2}\longrightarrow Aut\mathbb{Z}_{n}$. So for a fixed homomorphism, what can we say about $H^{2}(\mathbb{Z}_{2},\mathbb{Z}_{n})$? And how many actions can we fix?
By $\mathbb{Z}_{n}$ I mean the cyclic group and not a $p$-adic structure.