On finite group scheme-theoretical categories, II

Let $\mathcal{C}:=\mathcal{C}(G,\omega,H,\psi)$ be a finite group scheme-theoretical category over an algebraically closed field of characteristic $p\ge 0$ as defined by the first author. For any indecomposable exact module category over $\mathcal{C}$, we classify its simple objects and provide an expression for their projective covers in terms of double cosets and projective representations of certain closed subgroup schemes of $G$. This upgrades a result of Ostrik for group-theoretical fusion categories in characteristic $0$, and generalizes our previous work for the case $\omega=1$. As a byproduct, we describe the simples and indecomposable projectives of $\mathcal{C}$. Finally, we apply our results to describe the blocks of the center of ${\rm Coh}(G,\omega)$.