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In this last article of the series on outer actions of a countable dicreteamenable group on AFD factors, we analyze outer actions of a countable discretefree abelian group on an AFD factor of type $\text{III}_\lambda$, $0<\lambda<1$, and compute outer conjugacy invariants. As a byproduct, we discover theasymmetrization technique for coboundary condition on a $\Bbb T$-valued cocycleof a torsion free abelian group, which might have been known by the groupcohomologists. As the asymmetrization technique gives us a very handy criteriafor coboundaries, we present it here in detail in the second section.

Outer Actions of a Discrete Amenable Group on Approximately Finite  Dimensional Factors III, The Type $\text{III}_\lambda$ Case,  $\pmb{0<\lambda<1}$, Examples