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](https://mathoverflow.net/questions/58455/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!).



finite quotients of fundamental groups in positive characteristic
For a supersingular elliptic $E$ over an algebraically closed field of characteristic two or three there exists a smooth curve $C$ of higher genus such that $Aut\_0(E)$ is a finite quotient of $\pi\_1(C)$.


This is explained in section 3 of <http://arxiv.org/abs/1005.2142v3>


This is an easy application of a general theory of finite quotients of fundamental groups of smooth curves as explained in the paper


Amilcar Pacheco and Katherine F. Stevenson. Finite quotients of the algebraic fundamental group of projective curves in positive characteristic. Pacific J. Math., 192(1):143–158, 2000


In this paper, it is explained how to realize groups which have the property that their maximal $p$-Sylow subgroup ($p$ being the characteristic) is normal. The automorphism groups of supersingular elliptic curves satisfy this property.


