  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 ${\rm Coh}(G,\omega)$.
