Flat deformations of curves:: A local-global principal

Let C be a generically smooth, locally complete intersection curve defined over an algebraically closed field k of characteristic p >= 0. Let G subset of Aut(k) C be a finite group of automorphisms of C. We develop a theory of G-equivariant deformations of the Galois cover C -> C/G. We give a thorough study of the local obstructions, those localized at singular or widely ramified points, to deform equivariantly a cover. As an application, we discuss the case of G-equivariant deformations of semistable curves.