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](https://mathoverflow.net/questions/24754/cohomology-of-the-eilenberg-maclane-spaces-kg-n)), 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)?



Low degree cohomology of Eilenberg-MacLane space K(G,2)?
For a finite cyclic group G, in the range you ask for you get cohomology groups
$$\mathbb{Z}, 0, 0, G \cong Ext(G, \mathbb{Z}), 0.$$
One sees this by for example computing the Leray--Serre spectral sequence for
$$K(G, 1) \to \* \to K(G,2).$$


