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In this note, we examine the right action of the Steenrod algebra$\mathcal{A}$ on the homology groups $H_*(BV_s, \F_2)$, where $V_s = \F_2^s$.We find a relationship between the intersection of kernels of $Sq^{2^i}$ andthe intersection of images of $Sq^{2^{i+1}-1}$, which can be generalized toarbitrary right $\mathcal{A}$-modules. While it is easy to show that$\bigcap_{i=0}^{k} \mathrm{im}\,Sq^{2^{i+1}-1} \subseteq \bigcap_{i = 0}^k\mathrm{ker}\,Sq^{2^i}$ for any given $k \geq 0$, the reverse inclusion neednot be true. We develop the machinery of homotopy systems and null subspaces inorder to address the natural question of when the reverse inclusion can beexpected. In the second half of the paper, we discuss some counter-examples tothe reverse inclusion, for small values of $k$, that exist in $H_*(BV_s,\F_2)$.

Relations among the kernels and images of {S}teenrod squares acting on right $\mathcal{A}$-modules