  We show that certain twisting deformations of a family of supersolvable
groups are simple as Hopf algebras. These groups are direct products of two
generalized dihedral groups. Examples of this construction arise in dimensions
60 and p^2q^2, for prime numbers p, q with q dividing p-1. We also show that
certain twisting deformation of the symmetric group is simple as a Hopf
algebra. On the other hand, we prove that every twisting deformation of a
nilpotent group is semisolvable. We conclude that the notions of simplicity and
(semi)solvability of a semisimple Hopf algebra are not determined by its tensor
category of representations.
