Projectivity of a relative Hopf module over the subring of coinvariants

Let $k$ be a commutative ring, $H$ a faithfully flat Hopf algebra with bijective antipode, $A$ a $k$-flat right $H$-comodule algebra. We investigate when a relative Hopf module is projective over the subring of coinvariants $B=A^{\mathrm{c}\mathrm{o}H}$, and we study the semisimplicity of the category of relative Hopf modules.