Recall that for any space $X$, the cohomology $H^*X$ (always, in this post, with $\mathbb{Z}/2$-coefficients) has an action of the Steenrod algebra $\mathcal{A}$; that is, a natural morphism $\mathcal{A} \otimes H^*X \to H^*X$. This is not a morphism of algebras, but $\mathcal{A}$ has a Hopf algebra structure such that the appropriate diagrams involving comultilication on $\mathcal{A}$ and $H^*X$ commute.

Consider now a space with finitely generated homology in each degree. Then the Steenrod algebra acts on homology by duality, and dualizing this gives that the dual Steenrod algebra $\mathcal{A}^{\vee}$ coacts on the completed cohomology ring; that is, there is a map $$ H^*X \to H^*X \hat{\otimes} \mathcal{A}^{\vee}$$ which is in fact a morphism of rings, and makes appropriate diagrams commute for products in $X$. It follows that when $X$ is a H space, and $H^*X$ is a Hopf algebra, then we have a coaction of $\mathcal{A}^{\vee}$ on the completed cohomology ring $H^{\star \star}X$ (or something like that). Anyway, the upshot of this is that $\mathrm{Spec} \mathcal{A}^{\vee}$ is a (noncommutative) group scheme because of the Hopf algebra structure, and what we really have is an action of this group scheme (in some sense, at least) on the formal scheme $\mathrm{Sppf} H^{\star \star}(X)$.

In the case where $X = \mathbb{RP}^\infty$, then the formal scheme just described is the formal additive group. Milnor's paper "The Steenrod algebra and its dual" shows that $\mathrm{Spec} \mathcal{A}^{\vee}$ is precisely the automorphism group scheme of this (i.e. a polynomial ring on variables in each power of $2$ minus one). This is established by computation in Milnor's paper.

**Q1:** Is there a high-concept explanation of why this should the case?

**Q2:** What's the analog in characteristic $p \neq 2$?

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