  To study outer actions $\a$ of a group $G$ on a factor $\sM$ of type
{\threel}, $0<\la<1$, we study first the cohomology group of a group with the
unitary group of an abelian {\vna} as a coefficient group and establish a
technique to reduce the coefficient group to the torus $\T$ by the Shapiro
mechanism based on the groupoid approach. We then show a functorial
construction of outer actions of a {\cdag} on an AFD factor of type {\threel},
sharpening the result in \cite{KtT2: \S4}. The periodicity of the flow of
weights on a factor $\sM$ of type {\threel} allows us to introduce an
equivariant commutative square directly related to the discrete core. But this
makes it necessary to introduce an enlarged group $\Aut(\sM)_\txm$ relative to
the modulus homomorphism $\txm=\mod: \Aut(\sM)\mapsto \rt'z$. We then discuss
the reduced {\mhjr}, which allows us to describe the invariant of outer action
$\a$ in a simpler form than the one for a general AFD factor: for example, the
cohomology group $\tH_{\txm, \fs}^\out(G, N, \T)$ of modular obstructions is a
compact abelian group. Making use of these reductions, we prove the
classification result of outer actions of $G$ on an {\AFD} factor $\sM$ of type
\threel.
