For affine smooth curves over $k=\bar{k}$ of char. $p,$ Abhyankar's conjecture (proved by Raynaud and Harbater) tells us exactly which finite groups can be realized as quotients of their fundamental groups.

What about complete smooth curves, or more generally higher dimensional varieties? Are there results or conjectural criteria (or necessary conditions) for finite quotients of their $\pi_1?$ (Definitely, not too much was known around 1990; see Serre's Bourbaki article on this.)

In particular, let $G$ be the automorphism group of the supersingular elliptic curve in char. $p=2$ or $3$ (see supersingular elliptic curve in char. 2 or 3 for various descriptions of its structure). Is there (and if yes, how to construct) a projective smooth variety in char. $p$ having $G$ as a quotient of its $\pi_1?$ Certainly there are lots of affine smooth curves with this property (e.g. $\mathbb G_m$), and I wonder if for some of them, the covering is unramified at infinity (so that we win!).