This is a repost of my question here.
Do we know the structure of the étale fundamental group $\pi^\text{et}_1(\mathbb{G}_{m,K^\text{sep}})$ of the multiplicative group, for a given field $K$? For instance what is $$\pi_1^\text{et}(\mathbb{G}_{m,\overline{\mathbb{F}_p}})?$$
I am not sure if much is known about the structure of $\pi_1(\mathbf{G}_m)$ itself, but at least there is a complete characterization of its finite quotients. This is the content of the so-called Abhyankar's Conjecture proven by Raynaud and Harbater.
To formulate the conjecture, let me recall the definition of $p(G)$ for a group $G$. It is defined as the minimal subgroup of $G$ generated by all Sylow $p$-subgroup. It is easily seen that $p(G)$ is normal in $G$, so it makes sense to speak of the quotient group $G/p(G)$.
Abhyankar's Conjecture: Let $X$ be a smooth connected curve over an algebraically closed field $k$ of characteristic $p$. Suppose that $X$ is of genus $g$ and $S=\{s_0, \dots, s_r\}$ is a finite set of closed points of $X$. Then a finite group $G$ can be realized as a quotient of $\pi_1(X-S)$ if and only if $G/p(G)$ is generated by at most $2g+r$ elements.
The idea of Raynaud's proof for $X=\mathbf{P}^1$ is explained in Chapter $9$ of the book `Rigid analytic geometry and its applications' by Fresnel and van der Put. The general case is proven in the paper ``Abhyankar's conjecture on Galois groups over curves'' by Harbater.
Remark: There is a natural question if $\pi_1(X)$ can be reconstructed from the set of its finite quotients. It turns out to be false as $\pi_1(\mathbf{A}^n)$ and $\pi_1(\mathbf{A}^m)$ have the same finite quotients, but they have different cohomological dimensions. Look at Proposition $7.3.1$ in Achinger's paper for more details.
