Extensions of Finite Quantum Groups by Finite Groups

We give a necessary and sufficient condition for two Hopf algebras presented as central extensions to be isomorphic, in a suitable setting. We then study the question of isomorphism between the Hopf algebras constructed in [AG] as quantum subgroups of quantum groups at roots of 1. Finally, we apply the first general result to show the existence of infinitely many non-isomorphic Hopf algebras of the same dimension, presented as extensions of finite quantum groups by finite groups.