Periodic and strongly free automorphisms on inclusions of type IIIλ factors

We prove that a periodic and strongly free automorphism acting on a strongly amenable inclusion Q ⊂ P of type IIIλ factors, λ ∈ (0, 1), with a common discrete decomposition, is classified, up to conjugacy, by its period, its standard invariant on the type II core of Q ⊂ P, and its Connes-Takesaki module