Steenrod closed $C_3$-invariant parameter ideals in the mod 2 cohomology of $\mathbb{Z}\mathrm{/}2\times \mathbb{Z}\mathrm{/}2$

For the nontrivial action by the cyclic group $C_3$ of order $3$ on the graded polynomial ring ${\mathbb{F}}_2[a,b]$, we classify the $C_3$-invariant parameter ideals that are closed under Steenrod operations. The classification has applications to free actions by the Klein four-group $\mathbb{Z}\mathrm{/}2\times \mathbb{Z}\mathrm{/}2$ on products of two spheres (and more generally, finite CW complexes with four-dimensional mod $2$ homology) that extend to actions by the alternating group $A_4=(\mathbb{Z}\mathrm{/}2\times \mathbb{Z}\mathrm{/}2)⋊C_3$.