  We address the question whether the condition on a fusion category being
solvable or not is determined by its fusion rules. We prove that the answer is
affirmative for some families of non-solvable examples arising from
representations of semisimple Hopf algebras associated to exact factorizations
of the symmetric and alternating groups. In the context of spherical fusion
categories, we also consider the invariant provided by the $S$-matrix of the
Drinfeld center and show that this invariant does determine the solvability of
a fusion category provided it is group-theoretical.
