In An introduction to $\mathbb A^1$-homotopy theory ([1]) and On the motivic $\pi_0$ of the sphere spectrum ([2]) Morel describes a computation of $\bigoplus_{n\in \mathbb Z} [S^0, \mathbb G_m^{\wedge n}]$, where the brackets mean "homotopy classes of maps in $\mathbb P^1$-spectra over a perfect field $k$".
In [2] there isn't much more to that statement than that, but in [1] he gives the slightly more general motivation of trying to compute $[S^i,\mathbb G_m^{\wedge n}]$ for arbitrary $i,n$. Of course, he mentions that for $i<0$ this is simply $0$, and the theorem he describes is the $i=0$ case. But there seems to be no mention of the $i>0$ case.
One "obvious" reason for that to be complicated is that for $i>0$ and $n=0$, that would presumably be related to computing the stable homotopy groups of spheres (e.g. if $k$ can be embedded in $\mathbb R$, then I think that the classical stable homotopy groups of $S^0$ split off the ones in $\mathbb P^1$-spectra) - which is not a problem in the $i=0$ case (or $i\leq 0$ more generally).
But can we imagine a computation of these groups in terms of the classical stable homotopy groups of spheres ? We have a morphism from the graded ring $\pi_*(S^0)$ to the bigraded ring $\pi_{*,*}^{\mathbb P^1}(S^0)$ given by the (unique) symmetric monoidal colimit preserving functor $\mathbf{Sp}\to \mathcal{SH}^{\mathbb P^1}(k)$, so we can definitely view the latter as an algebra or a module over the former :
Is there a way to describe this algebra/module structure "explicitly" ?
By "explicitly" I mean a description similar to Morel's description of the $i=0$ case, where we have an expression in terms of Milnor-Witt groups. Speaking of which, I happen to have a reference question about the same thing, so if it's appropriate, I'll just ask it here too :
Are there other references than [1] and [2] where the computation of $[S^0,\mathbb G_m^{\wedge n}]$ is explained ? Ideally, more "expository" references.
