Given an action of a finite group G on a fusion category C we give a
criterion for the category of G-equivariant objects in C to be
group-theoretical, i.e., to be categorically Morita equivalent to a category of
group-graded vector spaces. We use this criterion to answer affirmatively the
question about existence of non group-theoretical semisimple Hopf algebras
asked by P. Etingof, V. Ostrik, and the author in math/0203060. Namely, we show
that certain Z/2Z-equivariantizations of fusion categories constructed by D.
Tambara and S. Yamagami are equivalent to representation categories of non
group-theoretical semisimple Hopf algebras. We describe these Hopf algebras as
extensions and show that they are upper and lower semisolvable.
