I saw a recent answer mention that one can think of monoids with zero as “$\mathbb{F}_{1}$-algebras”, since the former are monoids in the symmetric monoidal category of pointed sets (“$\mathbb{F}_1$-modules”) equipped with the smash product.

Now, the $\infty$-categorical analogue of commutative monoids and abelian groups are $\mathbb{E}_{\infty}$-spaces and spectra, the $\mathbb{E}_{\infty}$-monoids and $\mathbb{E}_{\infty}$-groups in the symmetric monoidal $\infty$-category $(\mathcal{S},\times,\mathrm{pt})$ of spaces.

Question. What are $\mathbb{E}_{\infty}$-monoids and $\mathbb{E}_\infty$-groups in the symmetric monoidal $\infty$-category $(\mathcal{S}_{*},\wedge,S^{0})$ of pointed spaces?

(Which I guess one may call "derived $\mathbb{F}_{1}$-algebras" in the terminology of the answer I mentioned above.)

Moreover, have these been studied before (under a less fancy name)?