Since the sphere spectrum is connective, we may view it as an $\mathbb{E}_{\infty}$-group in spaces, namely $QS^0$. This space, as any simplicial set, has a homotopy category $\mathsf{Ho}(QS^0)$, and since $\mathsf{Ho}$ is symmetric monoidal, $\mathsf{Ho}(QS^0)$ acquires the structure of an $\mathbb{E}_{\infty}$-group in categories, making it into a "grouplike" symmetric monoidal category (more commonly called a symmetric groupoidal category or an abelian $2$-group).

**Question.** Is there a known explicit description of the symmetric groupoidal category $\mathsf{Ho}(\mathbb{S})\overset{\mathrm{def}}{=}\mathsf{Ho}(QS^0)$?