Let $G \curvearrowright (X,\nu)$ be probability measure preserving action of a countable discrete group. When does there exist a probability measure preserving action $G \curvearrowright (Y,\mu)$ such that $G \curvearrowright (X,\nu)$ is isomorphic to the diagonal action $G \curvearrowright (Y \times Y, \mu \times \mu)$?

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