$$ \newcommand{\Z}{\mathbb{Z}} $$

Consider the group $G=\Z_2\ltimes B^2\Z^2$ where $\Z_2$ acts on $\Z^2$ by interchanging the two factors of $\Z^2$, hence $\Z_2$ also acts on the Eilenberg-MacLane space $B^2\Z^2=K(\Z^2,2)$.

Question: what is the mod 2 cohomology ring $H^*(BG,\Z_2)$ and its module structure over the mod 2 Steenrod algebra?

Here is my attempt (xymatrix failed to compile): There is a fibration $$\xymatrix{B^3\Z^2\ar[r]&BG\ar[d]\\&B\Z_2.}$$

So we have the Serre spectral sequence $$H^p(B\Z_2,H^q(B^3\Z^2,\Z_2))\Rightarrow H^{p+q}(BG,\Z_2).$$