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