Descent of Hilbert $C$-modules

Let $F$ be a right Hilbert $C$*-module over a $C$*-algebra $B$, and suppose that $F$ is equipped with a left action, by compact operators, of a second $C$*-algebra $A$. Tensor product with $F$ gives a functor from Hilbert $C$*-modules over $A$ to Hilbert $C$*-modules over $B$. We prove that under certain conditions (which are always satisfied if, for instance, $A$ is nuclear), the image of this functor can be described in terms of coactions of a certain coalgebra canonically associated to $F$. We then discuss several examples that fit into this framework: parabolic induction of tempered group representations; Hermitian connections on Hilbert $C$*-modules; Fourier (co)algebras of compact groups; and the maximal $C$*-dilation of operator modules over non-self-adjoint operator algebras.