@DenisNardin That's okay! In this case the most natural constraint is to have the $E_\infty$-structure life the cup product structure on cohomology. Also, I know very little about obstruction theory other than the basics. Any suggestions/references are welcome!

@DenisNardin Is it a bad question to ask then how many different $E_\infty$-structures exist?

@DylanWilson Writing down the structure maps for an operad interaction would certainly be interesting. Really my goal is to bring some of this $E_\infty$-language down to earth. Thanks for the suggestion by specializing in the case $G=\mathbb{Z}/p$!

@TylerLawson I'm not sure how explicity of a description you want for $C^*(K(G,n);\mathbb{Z})$. For example, I could take the singular cochains for the symmetric space of a moore space, $Sym^\infty(M(G,n))$. Also, I was not aware that there were competing definitions for $E_\infty$-algebras. I was using the definition in the Lurie-Gaitsgory paper on the Weil-conjectures for function fields.

@SaalHardali do you have a reference for this claim in the $A_\infty$ case?

Complex projective manifolds are always algebraic by the Kodaira embedding theorem. Also, you should not get excited about a smooth manifold embedding into a variety. By the whitney embed into some $\mathbb{R}^n$, and so if your manifold is compact, you can embed it into a neighborhood of a smooth point of some variety.

Also, why can the Hirzebruch surface have such a simpler presentation, as seen here: mathoverflow.net/questions/122952/on-a-hirzebruch-surface ?

K as in K-injective resolution.

I created a bounty for a question related to my comment: math.stackexchange.com/questions/2364401/…

This paper web.archive.org/web/20170815173103/http://www.math.uiuc.edu/… gives an excellent description for computing the steenrod squares on $K(\mathbb{Z}/2;k)$ starting with $\mathbb{RP}^\infty$.

Why do the differentials for the AHSS have to be a stable cohomology operation? Do you have a reference?

@DylanWilson Oh, apparently this is the case. Thanks for pointing this out!