  Let $\mathscr {C}(G,H,\psi)$ be a finite group scheme-theoretical category
over an algebraically closed field of characteristic $p\ge 0$ as introduced by
the first author. For any indecomposable exact module category over $\mathscr
{C}(G,H,\psi)$, 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, which upgrades a result by
Ostrik for group-theoretical fusion categories. As a byproduct, we describe the
simples and indecomposable projectives of $\mathscr {C}(G,H,\psi)$, and
parametrize the Brauer-Piccard group of ${\rm Coh}(G)$ for any finite connected
group scheme $G$. Finally, we apply our results to describe the blocks of the
center of ${\rm Coh}(G)$.
