We investigate the right action of the mod $p$ Steenrod algebra $\mathcal{A}_p$ on the homology $H_*(L^{\wedge s},\Z_p)$ where $L=B\mathbb{Z}_p$ is the lens space. Following ideas of Ault and Singer we investigate the relation between intersection of kernels of the reduced powers $P^{p^i}$ and Bockstein element $\beta$ and the intersection of images of $P^{p^{i+1}-1}$ and of $\beta$. Namely one can check that $\bigcap_{i=0}^k\operatorname{im}P^{p^{i+1}-1}\subset\bigcap_{i=0}^k\ker P^{p^i}$ and $\bigcap_{i=0}^k\operatorname{im}P^{p^{i+1}-1}\cap\,\operatorname{im}\beta\subset\bigcap_{i=0}^k\ker P^{p^i}\cap\,\ker\beta$. We generalize Ault's homotopy systems to $p>2$ and examine when the reverse inclusions are true.