Let $n=p^k$ be a prime power.

When $k=1$, the algebra of stable operations in mod $p$ cohomology is the Steenrod algebra $\mathcal{A}_p$. It has a nice description in terms of generators and relations. Its dual (as a Hopf algebra) is also well understood by work of Milnor.

What about when $k>1$? Has any work been done on trying to understand the algebra of stable operations in mod $n$ cohomology? For instance, is there a description of $\mathcal{A}_4$ in terms of generators and relations? What about the dual Hopf algebra?

Of course I am just asking about the cohomology ring $H^\ast (H\mathbb{Z}/n;\mathbb{Z}/n)$, where $H\mathbb{Z}/n$ denotes the mod $n$ Eilenberg--Mac Lane spectrum. So I fully expect the answer to be either one of "yes, this is well-known and classical" or "no, this is known to be a big mess".