I saw a recent [answer](https://mathoverflow.net/a/401329) mention that one can think of [monoids with zero](https://en.wikipedia.org/wiki/Absorbing_element) 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.

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**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)?