$p$-group Galois covers of curves in characteristic $p$. II

Let k be an algebraically closed field of characteristic p>0 and let G be a finite p -group. The results of Harbater, Katz and Gabber associate to every k -linear action of G on k\llbracket t\rrbracket an HKG-cover, i.e. a G -cover of the projective line ramified only over \infty . In this paper we relate the HKG-covers to the classical problem of determining the equivariant structure of cohomologies of a curve with an action of G . To this end, we present a new way of computing cohomologies of HKG-covers. As an application of our results, we compute the equivariant structure of the de Rham cohomology of Klein four covers in characteristic 2 .