Low-degree integral homology of a mod-two Eilenberg–MacLane space ($H_{n,n+2,n+3}(K(\mathbb{Z}/2,n);\mathbb{Z})=\mathbb{Z}/2(n\ge 4)$)

For $n\ge 4$, the first four possible degrees have integral groups $\mathbb{Z}/2,0,\mathbb{Z}/2,\mathbb{Z}/2$ in degrees $n,n+1,n+2,n+3$. Mod-two Betti numbers count cyclic summands, while the nonzero first Bockstein actions on $u,{\mathrm{Sq}}^{2}u,{\mathrm{Sq}}^2{\mathrm{Sq}}^{1}u$ show that their orders are two. This prevents mistaking a nonzero mod-two group for a higher-order integral cyclic group.