Raynaud’s group-scheme and reduction of coverings

Let Y K → X K be a Galois covering of smooth curves over a field of characteristic 0, with Galois group G. We assume K is the fraction field of a discrete valuation ring R with residue characteristic p. Assuming p 2 ∤ G and the p-Sylow subgroup of G is normal, we consider the possible reductions of the covering modulo p. In our main theorem we show the existence, after base change, of a twisted curve $$\mathcal{X} \rightarrow Spec (R)$$ , a group scheme $$\mathcal{G}\rightarrow \mathcal{X}$$ and a covering $$Y \rightarrow \mathcal{X}$$ extending Y K → X K , with Y a stable curve, such that Y is a $$\mathcal{G}$$ -torsor.In case p 2 | G counterexamples to the analogous statement are given; in the appendix a strong counterexample is given, where a non-free effective action of α p 2 on a smooth 1-dimensional formal group is shown to lift to characteristic 0.