This is motivated by the discussion here. The usual definition of the etale fundamental group (as in SGA 1) gives automatic profiniteness: Grothendieck's formulation of Galois theory states that any category with a "fiber functor" to the category of finite sets satisfying appropriate hypotheses is isomorphic to the category of finite, continuous $G$-sets for a well-defined profinite group $G$ (taken as the limit of the automorphism groups of Galois objects, or as the automorphism group of said fiber functor). In SGA1, the strategy is to take finite etale covers of a fixed (connected) scheme with the fiber functor the set of liftings of a geometric point.

One problem with this approach is that $H^1(X_{et}, G)$ for a finite group $G$ (which classifies $G$-torsors in the etale topology) is not isomorphic to $\hom(\pi_1(X, \overline{x}), G)$ unless $G$ is finite. Indeed, this would imply that the cohomology of the constant sheaf $\mathbb{Z}$ would always be trivial, but this is not true (e.g. for a nodal cubic). However, Scott Carnahan asserts on the aforementioned thread that the "right" etale fundamental group of a nodal cubic should be $\mathbb{Z}$, not something profinite.

How exactly does this work? People have suggested that one can define it as a similar inverse limit of automorphism groups, but is there a similar equivalence of categories and an analogous formalism for weaker "Galois categories"? (Perhaps one wants not just all etale morphisms but, say, torsors: the disjoint union of two open immersions might not be the right candidate.) I'm pretty sure that the finiteness is necessary in the usual proofs of Galois theory, but maybe there's something more.