Geodesic
Outer conjugacy of the actions of countable amenable groups on a~measure space
Izvestiya. Mathematics , Tome 29 (1987) no. 1, pp. 1-18.

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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. 
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S. I. Bezuglyi; V. Ya. Golodets. Outer conjugacy of the actions of countable amenable groups on a~measure space. Izvestiya. Mathematics , Tome 29 (1987) no. 1, pp. 1-18. http://geodesic.mathdoc.fr/item/IM2_1987_29_1_a0/

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