  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 $Sq^{2^i}$ and
the intersection of images of $Sq^{2^{i+1}-1}$, which can be generalized to
arbitrary 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 need
not be true. We develop the machinery of homotopy systems and null subspaces in
order to address the natural question of when the reverse inclusion can be
expected. In the second half of the paper, we discuss some counter-examples to
the reverse inclusion, for small values of $k$, that exist in $H_*(BV_s,
\F_2)$.
