I'm currently attempting to understand Galois theory for schemes, largely following the books *Galois Theory for Schemes* by Henrik Lenstra and *Galois Groups and Fundamental Groups* by Tamas Szamuely. The main theorem is

Let $X$ be a connected scheme. Then there exists a profinite group $\pi$, uniquely determined up to isomorphism, such that the category $\mathbf{FEt}_X$ of finite \'etale covers of $X$ is equivalent to the category $\pi$-$\mathbf{sets}$ of finite sets on which $\pi$ acts continuously.

I'm attempting to come up with some explicit examples of this equivalence/correspondence for specific schemes. If $X = \operatorname{Spec} k$ for some field $k$, then one simply recovers (Grothendieck's formulation of) Galois theory for fields. If $X = \operatorname{Spec} \mathbf{Z}$, then the finite \'etale covers just look like disjoint unions of $\operatorname{Spec} \mathbf{Z}$, and the profinite group $\pi$ is trivial.

Both of these examples seem pretty boring though. Does anyone know of some interesting examples classifying the finite \'etale covers of some other scheme?