Let $H$ be a semisimple Hopf algebra over an algebraically closed field of characteristic zero. Further, let $K\subseteq H$ be a normal Hopf subalgebra. As we all know, $H$ then can be reconstructed from $K$ by some compatible data .
I want to know if there exists a similar result on Rep($H$) and Rep($K$), where Rep($H$) is the fusion category of finite-dimensional representations of $H$? If not, what can be said about Rep($H$) and Rep($K$)?
N. Andruskiewitsch, Notes on extensions of Hopf algebras, Canad. J. Math. 48 (1996), 3-42