Wild Hurwitz spaces and level structures

Hurwitz moduli spaces of covers of curves of degree d are classical and well studied objects if one assumes that d! is invertible and hence no wild ramification phenomena occur.
There were very few attempts to study the wild case. In the most important one Abramovich and Oort started with the classical space $H_{2,1,0,4}$ of double covers of P^1 ramified at four points and (following an idea of Kontsevich and Pandariphande) described its schematic closure H in the space of stable maps over Z. The result over $F_2$ was both strange and informative, but lacked a modular interpretation.