Recall that an Eilenberg-Maclane space $K(G, n)$ is characterized by $\pi_i(K(G,n)) = G$ if $i=n$ and is trivial otherwise. (Of course $G$ should be abelian if $n>1$.)

I'm aware that computing $H^j(K(G,n), \mathbb Z)$ for general $j$ and $n$ is not so easy (see, e.g., here), but I'm hoping that for certain small values of $j$ and $n$ it's easier.

My question: Is there a good reference for $H^j(K(G,2), \mathbb Z)$, where $j \le 4$ and $G$ is finite abelian (or just cyclic)?