We can characterise $\mathbb{Z}$ and $\mathbb{Z}/2$ as the corepresenting abelian groups of the functors \begin{align*} (-)^\times &\colon \mathsf{Ab} \to \mathsf{Sets},\\ \mathrm{Inv} &\colon \mathsf{Ab} \to \mathsf{Sets} \end{align*} given by $A\mapsto A^\times$ and $A\mapsto\mathrm{Inv}(A)\overset{\mathrm{def}}{=}\left\{a\in A\ \middle|\ a^2=1_A\right\}$.

A similar approach in the $\infty$-world gives the sphere spectrum $\mathbb{S}$ and $``\mathbb{S}/2\text{''}\overset{\mathrm{def}}{=}\Omega Q\mathbb{RP}^\infty$. Here are the first $8$ homotopy groups of $\mathbb{S}$ and $\mathbb{S}/2$, for comparison: $$ \begin{aligned} \pi_0(\mathbb{S}) &\cong \mathbb{Z},\\ \pi_1(\mathbb{S}) &\cong \mathbb{Z}/2,\\ \pi_2(\mathbb{S}) &\cong \mathbb{Z}/2,\\ \pi_3(\mathbb{S}) &\cong \mathbb{Z}/24,\\ \pi_4(\mathbb{S}) &\cong 0,\\ \pi_5(\mathbb{S}) &\cong 0,\\ \pi_6(\mathbb{S}) &\cong \mathbb{Z}/2,\\ \pi_7(\mathbb{S}) &\cong \mathbb{Z}/16\times\mathbb{Z}/3\times\mathbb{Z}/5, \end{aligned} \quad\quad \begin{aligned} \pi_0(\mathbb{S}/2) &\cong \mathbb{Z}/2,\\ \pi_1(\mathbb{S}/2) &\cong \mathbb{Z}/2,\\ \pi_2(\mathbb{S}/2) &\cong \mathbb{Z}/8,\\ \pi_3(\mathbb{S}/2) &\cong \mathbb{Z}/2,\\ \pi_4(\mathbb{S}/2) &\cong 0,\\ \pi_5(\mathbb{S}/2) &\cong \mathbb{Z}/2,\\ \pi_6(\mathbb{S}/2) &\cong \mathbb{Z}/16\times\mathbb{Z}/2,\\ \pi_7(\mathbb{S}/2) &\cong \mathbb{Z}/2\times\mathbb{Z}/2\times\mathbb{Z}/2. \end{aligned} $$ (The ones for $\mathbb{S}/2$ are taken from Liulevicius; see also MO 230790.)

What (homotopy associative, homotopy commutative, $\mathbb{A}_k$-, $\mathbb{E}_k$-, or $\mathbb{E}_\infty$-) ring spectra structures, if any, are there on $\mathbb{S}/2$?