A note on intermediate subfactors of Krishnan-Sunder subfactors

A Krishnan-Sunder subfactor $R_U \ss R$ of index $k^2$ is constructed from a permutation biunitary matrix $U\in M_p(\mathbb{C})\otimes M_k(\mathbb{C})$, i.e. the entries of $U$ are either 0 or 1 and both $U$ and its block transpose are unitary. The author previously showed that every irreducible Krishnan-Sunder subfactor has an intermediate subfactor by exhibiting the associated Bisch projection. The author has also shown in a separate paper that the principal and dual graphs of the intermediate subfactor are the same as those of the subfactor $R^{\grp} \ss R^{H}$, where $H\ss \grp$ is an inclusion of finite groups with an outer action on $R$. In this paper we give a direct proof that the intermediate subfactor is isomorphic to $R^{\grp} \ss R^{H}$.