Let H be a finite-dimensional and semisimple Hopf algebra over an algebraically closed field of characteristic 0 such that H has exactly one isomorphism class of simple modules that have not dimension 1. These Hopf algebras were the object of study in, for instance, 1 and 9. In this paper we study this property in the context of certain abelian extensions of group algebras and give a group theoretical criterion for such Hopf algebras to be of the above type. We also give a classification result in a special case thereof.
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