Is there a known explicit description of the [abelian $2$-group](https://en.wikipedia.org/wiki/Abelian_2-group) $\mathsf{Ho}(\mathbb{S})\overset{\mathrm{def}}{=}\mathsf{Ho}(QS^0)\cong\Pi\_{\leq1}(QS^0)$?



What is the homotopy category of the sphere spectrum?
This is the groupoid given by the 1-truncation $\tau\_{\leq 1}(QS^0)$. This groupoid has $\mathbb Z$-many objects (since $\pi\_0^s = \mathbb Z$), and each one has automorphism group $C\_2$ (since $\pi\_1^s = C\_2$). The tensor product on objects is given by addition in $\mathbb Z$, and on morphisms by addition in $C\_2$. One way to see this is to consider the universal functor $\Sigma \to QS^0$ given by the Barratt-Priddy-Quillen theorem (i.e. the fact that $K(\Sigma) = QS^0$; here $\Sigma$ is the groupoid of finite sets with the disjoint union monoidal structure), and to postcompose with the truncation functor $QS^0 \to \tau\_{\leq 1} (QS^0)$; the fact that this functor is symmetric monoidal yields this description of the category. This perspective is discussed a bit more [here](https://mathoverflow.net/questions/386111/is-every-otimes-invertible-object-coherently-sym-central).


From the description I've given, I suppose it follows that $\tau\_{\leq 1} (QS^0)$ splits  symmetric  monoidally as $\tau\_{\leq 1} (QS^0) = \mathbb Z \times BC\_2$, (where $\mathbb Z$ is a discrete symmetric monoidal groupoid and $BC\_2$ is a 1-object symmetric monoidal groupoid), which is maybe a little surprising. This is not to say that $\tau\_{\leq 1} \mathbb S$ splits...


