Indicators of Bismash Products from Exact Symmetric Group Factorizations

We prove that all irreducible representations of the bismash product $H = $\mathbb{k}$ ^G # $\mathbb{k}$ S_{n-k}$ have Frobenius-Schur indicators +1 or 0 where $\mathbb{k}$ is an algebraically closed field and $S_n=S_{n-k}\cdot G$ is an exact factorization. Moreover, we have an description of those simple modules which are not self-dual. We prove some new results for bismash products in general and make conjectures about bismash products that arise from other exact factorizations of $S_n$.