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.

If the answer turns out to be no, then I would like to know which profinite groups arise as fundamental groups.

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.