Commutativity of automorphisms of subfactors modulo inner automorphisms

We introduce a new algebraic invariant chi(a)(M, N) of a subfactor N subset of 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 Out(M, N) in many interesting examples such as quantum SU(n)(k) subfactors with level k (k greater than or equal to 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.