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 \cite{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.