Let $(N \subset M)$ be a finite index inclusion of ${\rm II}_1$ factors, and $N \subset M \subset M_1$ the basic construction.  
The subfactor  $(N \subset M)$ is called self-dual if it is isomorphic to its dual $(M \subset M_1)$.  

Let $R$ be the hyperfinite ${\rm II}_1$ factor, let $G$ be a finite abelian group and $H$ a subgroup of $Aut(G)$.    

*Questions*: Is the subfactor  $(R ^ {G \rtimes H} \subset R^H)$ self-dual?     
Is every self-dual group-subgroup subfactor of this form?  

[ We use *the* outer action of any finite group $K$ on $R$, and the fixed point subfactor $R^K \subset R$ ]