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. We will use the outer action of any finite group $G$ on $R$, and the fixed point subfactor $R^G \subset R$.

Remark: Let $G$ be a finite group. Then, $(R ^ {G} \subset R)$ is self-dual iff $G$ is self-dual, iff $G$ is abelian.

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?