On the conjugacy of measurable partitions with respect to the normalizer of a full type $\mathrm{II}_1$ ergodic group

Let $G$ be a countable ergodic group of automorphisms of a measure space $(X,\mu)$ and $\mathcal{N}[G]$ be the normalizer of its full group $[G]$. Problem: for a pair of measurable partitions $\xi$ and $\eta$ of the space $X$, when does there exist an element $g\in\mathcal{N}[G]$ such that $g\xi=\eta$? For a wide class of measurable partitions, we give a solution to this problem in the case where $G$ is an approximately finite group with finite invariant measure. As a consequence, we obtain results concerning the conjugacy of the commutative subalgebras that correspond to $\xi$ and $\eta$ in the type $\mathrm{II}_1$ factor constructed via the orbit partition of the group $G$.