By works of Sutherland-Takesaki and Kawahigashi-Sutherland-Takesaki, actions of a discrete amenable group on an approximately finite- dimensional factor of type III have been classified up to cocycle conjugacy in terms of four invariants: a certain normal subgroup, the module, the characteristic invariant and the modular invariant.
For a given action of an abelian group on a von Neumann algebra, since the dual action on the crossed product algebra is canonically defined, the invariants of the dual action should be completely determined by those of the original action. In this note, we shall compute the invariants of the dual action of an action of a finite abelian group in two cases, one is that the module is trivial and the other is that the normal subgroup is trivial. Similar calculus have been done in H. Kosaki and T. Sano [‘Non-splitting inclusions of factors of type \(\text{III}_0\)’ (preprint)] for \(\mathbb{Z}_2\)-case. In that case, one of the above assumptions is automatically satisfied.