Equivariantly normal varieties for diagonalizable group actions

Given a finite group scheme G over a field and a G-variety X, we obtain a criterion for X to be G-normal in the sense of {Br24}. When G is diagonalizable, we describe the local structure of G-normal varieties in codimension $1$ and their dualizing sheaf. As an application, we obtain a version of the Hurwitz formula for G-normal varieties, where G is linearly reductive.