Here is one of the motivations for my question, when $p=2$. The homology of the spectrum $H\mathbb F_2$ as an algebra is generated by the Dyer Lashof operations on the single generator $\xi_1$ (and it is enough to consider the $E_2$ operation), and for $H\mathbb Z/2^k$ ($k>1$), we have to take more generators, i.e. $x$ the dual of the higher Bockstein, $\xi_1^2$ and $\overline\xi_2$. A reason for that is that $x$ vanishes under the Dyer Lashof operations, because the higher bockstein vanishes under the Steenrod operations.

Somehow I have the feeling that looking at the mod $2$ homology of the spectra $H\mathbb Z/2^k$ gives very degenerate information (it doesn't even depend on $k$), and that some statement like the very informal following one should be true:

$H_*(H\mathbb Z/2^k;\mathbb Z/2^k)$ is generated as an algebra over the mod $2^k$ Dyer Lashof operations, by a single generator $\xi^{(k)}_1$

Here $\xi^{(k)}_1$ would be dual to the Bockstein associated to the exact sequence $$\mathbb Z/2^k\to \mathbb Z/2^{k+1}\to\mathbb Z/2^k$$ in some sense.

Can all that make sense ? Is there such a thing as mod $p^k$ Dyer Lashof operations ? Of course there is such a thing but I am wondering to which extent this notion is relevant and has been studied.

freeE_2-algebra, the same is not true for the Z/p^k-dual Steenrod algebra) $\endgroup$ – Dylan Wilson Nov 24 '20 at 20:16