Classification of actions of discrete amenable groups on strongly amenable subfactors of type III$\lambda$

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.