Question: Can you check a generic point of the family for what configuration space will work for it? I've had several projects about such fundamental groups. Here is the URL to one, which I regard as a generalization to wild ramification of half of Grothendieck's famous result on the fundamental group of curves. This says that if you have a tamely ramified cover of the sphere (in characteristic 0), then there is no obstruction to its deformation as a cover. That is, any local p-adic deformation of its branch points, gives a unique local p-adic deformation of the cover. Uniqueness is the key point. Another way to say that considers a connected family of tamely ramified covers branched at r (distinct points) of the sphere P^1. Then, such a family gives a natural map to the space U_r of r distinct (unordered points on the sphere). I call U_r a configuration map. The result would then be, in the etale topology the map to the configuration space and the cover at a given point, determine the family in the etale topology. The generalization is in the paper at the following location. That considers any connected family of wildly ramified covers that are also branched at r points of the sphere. It is important that we do not assume the covers are Galois. In fact, there is no trivial way to reduce to the Galois case in positive characteristic (as there is in characteristic 0; the paper explains why). The result produces a natural finite type space \sP that maps to U_r. Then, the result is the same: There is a natural map in the finite (rather than etale topology) of the family to \sP. Further, up to isotriviality (in the finite topology) the map to \sP together with the cover at one point, determines the family (in the finite topology). In the finite topology means, that two such families will be isomorphic after pullback to a finite cover of \sP. Understanding the result in more depth requires understanding the space \sP which generalizes the notion of higher ramification groups. On my home page http://math.uci.edu/~mfried, section: I.a. Articles and Talks: --> * Articles: R(egular)I(nverse)G(alois)P(roblem) and arithmetic of covers (outside Modular Towers) --> Item #16 with A. Mezard, Configuration spaces for wildly ramified covers, Arithmetic fundamental groups and noncommutative algebra, PSPUM vol. of the American Math. Society (2002), 223--247.