Commutativity of automorphisms of subfactors modulo inner automorphisms

We introduce a new algebraic invariant $\chi _{a}(M,N)$ of a subfactor $N \subset M$. We show that this is an abelian group and that if the subfactor is strongly amenable, then the group coincides with the relative Connes invariant $\chi (M,N)$ introduced by Y. Kawahigashi. We also show that this group is contained in the center of ${\hbox {Out}}(M,N)$ in many interesting examples such as quantum $SU(n)_{k}$ subfactors with level $k$ $(k \geq n+1)$, but not always contained in the center. We also discuss its relation to the most general setting of the orbifold construction for subfactors.