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.

1. What are the $\mathbb{E}_{\infty}$-monoids in the symmetric monoidal $\infty$-category $(\mathcal{S}_{*},\wedge,S^{0})$ of pointed spaces? (In a sense, these may be called "spectral $\mathbb{F}_{1}$-algebras")
2. Have these been studied before?
3. The symmetric monoidal $\infty$-categories of $\mathbb{E}_\infty$-spaces and spectra have point-set models, such as $\Gamma$-spaces, commutative monoids in $*$-modules, and commutative monoids in $\mathcal{I}$-spaces for $\mathbb{E}_\infty$-spaces; and $\mathbb{S}$-modules, symmetric spectra, and orthogonal spectra for spectra. Does the symmetric monoidal $\infty$-category $\mathsf{Alg}_{\mathbb{E}_{\infty}}(\mathcal{S}_*)$ admit point-set models?