TITLE: Profinite groups as étale fundamental groups
QUESTION [25 upvotes]: Does every profinite group arise as the étale fundamental group of a connected scheme?
Equivalently, does every Galois category arise as the category of finite étale covers of a connected scheme?
Not every profinite group is an absolute galois group of a field (the only finite ones have order $1$ or $2$ by Artin-Schreier). Therefore we cannot restrict to spectra of fields. Perhaps one first has to check if every finite group arises as a fundamental group of a scheme. I don't even know enough examples to answer this question for cyclic groups. At least order $3$ is possible (see here,  Remark 2).
If the answer turns out to be no, then I would like to know which profinite groups arise as fundamental groups.

REPLY [6 votes]: Every finitely presented group occurs as the fundamental group of an irreducible complex variety (see below), so (at least) you get the completions of all those.
Simpson, Carlos. Local systems on proper algebraic $V$-manifolds. Pure Appl. Math. Q. 7 (2011), no. 4, Special Issue: In memory of Eckart Viehweg, 1675--1759. MR2918179