  We calculate Frobenius-Schur indicator values for some fusion categories
obtained from inclusions of finite groups $H\subset G$, where more concretely
$G$ is symmetric or alternating, and $H$ is a symmetric, alternating or cyclic
group. Our work is strongly related to earlier results by
Kashina-Mason-Montgomery, Jedwab-Montgomery, and Timmer for bismash product
Hopf algebras obtained from exact factorizations of groups. We can generalize
some of their results, settle some open questions and offer shorter proofs;
this already pertains to the Hopf algebra case, while our results also cover
fusion categories not associated to Hopf algebras.
