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.

an E_2-structure gives actions of the homology of braid groups, right?

so the answer should have something to do with the image of the map H_{\bullet}(B_n, k) -> H_{\bullet}(S_n, k), maybe

@QiaochuYuan That tracks, conceptually. I guess I was hoping for a more explicit answer, though, which would come from understanding that map on homology.

presumably somebody knows all about the homology of the E_2 operad over a finite field, but that somebody is not me

This should be somewhere in May's "homology of iterated loop spaces"

probably around page 213 ("Operations on C_{n+1} spaces")

pages.iu.edu/~mmandell/talks/Austin3.pdf This seems to address the question.

there's going to be a formal announcement shortly, but applications are now open for Talbot 2015: math.mit.edu/conferences/talbot

@Saul: sweet!