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
$$\pi_1^\text{et}(\mathbb{G}_{m,\overline{\mathbb{F}_p}})?$$


  [here]: https://math.stackexchange.com/questions/3419967/etale-fundamental-group-of-multiplicative-group-over-an-algebraically-separabl