  Using the continuous decomposition, we classify strongly free actions of
discrete amenable groups on strongly amenable subfactors of type III$_\lambda,
0 < \lambda < 1$. Winsl{\o}w's fundamental homomorphism is a complete
invariant. This removes the extra assumptions in the classification theorems of
Loi and Winsl{\o}w and gives a complete classification up to cocycle conjugacy.
