Outer conjugacy of the actions of countable amenable groups on a measure space

The following assertion is proved. Let $T$ be an automorphism of a Lebesgue space $(X,\mu)$, preserving the (finite or infinite) measure $\mu$, and let $U_i(G)$, $i=1,2$, be actions of a countable amenable group $G$ by automorphisms on $(X,\mu)$, such that $U_i(G)\subset N[T]$, where $N[T]$ is the normalizer of the full group $[T]$. For the existence of an automorphism $\theta\in N[T]$ such that $U_1(g)=\theta^{-1}U_2(g)t\theta$ (the outer conjugacy of the actions $U_1$ and $U_2$), where $t=t(g)\in[T]$, $g\in G$, it is necessary and sufficient that
\begin{gather*} \{g\in G:U_1(g)\in[T]\}=\{g\in G:U_2(g)\in[T]\},\\ \frac{d\mu\circ U_1(g)}{d\mu}=\frac{d\mu\circ U_2(g)}{d\mu}\quad(g\in G). \end{gather*}
The proof uses properties of cocycles of approximable groups of automorphisms.
Bibliography: 25 titles.