  The coset $G$-space of a finite group and a subgroup is a fundamental module
of study of Schur and others around 1930; for example, its endomorphism algebra
is a Hecke algebra of double cosets. We study and review its generalization $Q$
to Hopf subalgebras, especially the tensor powers and similarity as modules
over a Hopf algebra, or what's the same, Morita equivalence of the endomorphism
algebras. We prove that $Q$ has a nonzero integral if and only if the modular
function restricts to the modular function of the Hopf subalgebra. We also
study and organize knowledge of $Q$ and its tensor powers in terms of
annihilator ideals, sigma categories, trace ideals, Burnside ring formulas, and
when considering semisimple Hopf algebras, the depth of $Q$ in terms of the
McKay quiver and the Green ring.
