Let $S$ be an integral noetherian regular scheme, and let $X =\mathbb{G}_{m,S}$.

How to compute $\pi_1^{et}(X)$?

Note. I am only interested in the part not coming trivially from the finite etale covers. For instance, if $S$ is the spectrum of the field of rational numbers, $\pi_1^{et}(X)$ surjects onto the Galois group of $\mathbb{Q}$.

If $S$ is the spectrum of an algebraically closed field of characteristic zero, this group can be computed over the field of complex numbers, and we get $\widehat{\mathbb{Z}}$ as the connected finite etale covers of $X$ are in this case $x\mapsto x^n$ (with $n>0$).

What if $S$ is a dense open of Spec $\mathbb Z$? What if $S$ is of finite type over Spec $\mathbb Z$? What if $S$ is a variety over $\mathbb F_p$?