Cohomology of $K(\mathbb{Z}_2, n)$

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Is it true, for example, that $H^5(K(\mathbb{Z}_2,2),\mathbb{Z})=\mathbb{Z}_4$, so these groups have not only 2-torsion? Has question about integral cohomology ring of $K(\mathbb{Z}_2, n)$ easy answer, or spectral sequences over $\mathbb{Z}$ are not as simple as over $\mathbb{Z}_2$?