This talk concerns a variant of the Schottky problem, which asks to classify Jacobians among all abelian varieties. In characteristic p, there is a rich extra structure to consider. Namely, in characteristic p, abelian varieties can be partitioned into so-called Ekedahl-Oort strata, within which all abelian varieties have isomorphic p-torsion group schemes. From this point of view, it is fruitful to investigate which p-torsion group schemes can occur as the p-torsion of the Jacobian of a (specific type of) curve. In this talk, we treat the 2-torsion of curves in characteristic 2 that admit a separable double cover to another curve. Through an analysis of the first De Rham cohomology, we prove that the p-torsion of a double cover of an ordinary curve is determined by the ramification breaks of the cover. This generalizes a result by Elkin and Pries, where the base curve is the projective line, so that the covers are hyperelliptic curves. When the base curve is not ordinary, we establish bounds on the Ekedahl-Oort type of the cover.