Invertible bimodule categories over the representation category of a Hopf algebra

For any finite-dimensional Hopf algebra H we construct a group homomorphism BiGal(H) → BrPic(Rep(H)), from the group of equivalence classes of H-biGalois objects to the group of equivalence classes of invertible exact Rep(H)-bimodule categories. We discuss the injectivity of this map. We exemplify in the case H = Tq is a Taft Hopf algebra and for this we classify all exact indecomposable Rep(Tq)bimodule categories.