Let $X$ be a space, and consider its cochain algebra $C^*(X; k)$ for some finite field $k$, maybe just $\mathbb{F}_3$ if you like. This is an $\mathbb{E}_\infty$-algebra over $k$, and the resulting action of the $k$-homology of symmetric groups on its cohomology is responsible for the Steenrod operations.
My question is, if I just remember the $\mathbb{E}_2$-structure on $C^*(X;k)$, which Steenrod operations persist to the cohomology?
More generally, what operations persist when I restrict to an $\mathbb{E}_n$-structure?
I guess what I actually care about is what kinds of operations are preserved if I look at $\mathbb{E}_n$-maps between $\mathbb{E}_\infty$-rings, and that's probably what I should have asked in the first place.
@QiaochuYuan That tracks, conceptually. I guess I was hoping for a more explicit answer, though, which would come from understanding that map on homology.