The title and tags say it all: I am looking for a clean statement and proof of the equivalence of categories between finite étale covers of a connected $k$-scheme and finite continuous permutation representations of the étale fundamental group $\pi_1(X, \overline{x})$ given by the fiber functor.

I need this result for my thesis, but I do not have the space necessary to prove it, so I would like a clear reference I can cite. This is stated in passing at the start of Section 6 in Szamuely's *Galois Group and Fundamental Groups*, but the proof consists of various remarks spread out throughout the entirety of the preceding section, and even then there's some work to be done. If possible, I would love a a wholly contained proof so that I can reference this away. I only need the result for reductive group schemes over the algebraic closure of a finite field, so total generality is not necessary.