Let $V$ be the Klein four group $\{1,\sigma,\tau,\sigma \tau\}$. Consider the $V$-module $A$ which is the cokernel of the map $\mathbb{Z} \to \mathbb{Z}[V]$ which sends $1$ to $1 + \sigma + \tau + \sigma \tau$, where we are considering the action of $V$ on $\mathbb{Z}[V]$ by translation. This induces an action of $V$ on $A$.
How to calculate the cohomology groups $H^i(V,A)$ for $i = 1,2,3$ for this action?
I thought about using the associated long exact sequence in cohomology, but I'm not sure which cohomology groups in this long exact sequence are zero and which aren't...