 Low-degree mod-two cohomology of an Eilenberg–MacLane space

For $n\ge 2$, the positive-degree dimensions in degrees $n,n+1,n+2,n+3$ are $1,1,1,2$. The first three classes are $u,{\mathrm{Sq}}^{1}u,{\mathrm{Sq}}^{2}u$. In degree $n+3$, use ${\mathrm{Sq}}^{3}u$ and ${\mathrm{Sq}}^2{\mathrm{Sq}}^{1}u$ for $n\ge 3$; at $n=2$ replace the unstable first class by $u{\mathrm{Sq}}^{1}u$. At $n=2$ the degree-four class is $u^2$, and at $n=3$ the degree-six class ${\mathrm{Sq}}^{3}u$ is $u^2$.