$\newcommand{\K}{\mathrm{K}}$The abelian group completion functor $\K_0\colon\mathsf{CMon}\to\mathsf{Ab}$ satisfies $$ \K_0(A) \cong \mathbb{Z}\otimes_{\mathbb{N}}A, $$ naturally in $A\in\mathrm{Obj}(\mathsf{CMon})$, where

- $\mathbb{Z}$ is the additive monoid of integers (i.e. $\K_0(\mathbb{N})$, the group completion of $\mathbb{N}$);
- $\otimes_\mathbb{N}$ is the tensor product of commutative monoids.

**Question.** Does the $\mathbb{E}_{\infty}$-group completion functor $\K_0\colon\mathsf{Mon}_{\mathbb{E}_\infty}(\mathcal{S})\to\mathsf{Grp}_{\mathbb{E}_\infty}(\mathcal{S})$ similarly satisfies
$$\K_0(X)\cong QS^{0}\otimes_\mathbb{F}X,$$
where now

- $QS^0$, the stabilization of $S^0$, is the $\mathbb{E}_{\infty}$-group completion of $\mathbb{F}=\coprod_{n\in\mathbb{N}}\mathbf{B}\Sigma_{n}$, the groupoid of finite sets and permutations;
- $\otimes_{\mathbb{F}}$ is the tensor product of $\mathbb{E}_{\infty}$-spaces?