Let $X$ be a set, $\mathcal{B}$ a $\sigma$-algebra on $X$ and $\mu$ a probability measure on $(X,\mathcal{B})$.
Suppose $\mathcal{C}$ is a $\sigma$-subalgebra of $\mathcal{B}$. Then, the Loomis-Sikorski representation theorem tells us that $\mathcal{C}$ is actually isomorphic to the quotient of the $\sigma$-algebra $\mathcal{N}$ of some measure space $(Y,\mathcal{N},\mu')$ by the $\sigma$-ideal of $\mu'$-negligible sets.
Now suppose some countable group $G$ acts on $X$ in a measure preserving way, that is, for every $g \in G$, the function $X \to X$ given by $x \mapsto gx$ is a Borel isomorphism and $\mu(gA) = \mu(A)$ for every $A \in \mathcal{B}$.
Is there an equivariant version of the Loomis-Sikorski-Stone representation theorem? I'm wondering whether the following statement is true:
If $\mathcal{C}$ is a $\sigma$-subalgebra of $\mathcal{B}$ closed under the action of $G$, then there exists some measure space $(Y,\mathcal{N},\mu')$ and some measure preserving action of $G$ on $Y$ such that $\mathcal{C}$ is $G$-isomorphic to the quotient of $\mathcal{N}$ by the $\sigma$-ideal of $\mu'$-negligible sets.
A reference, tip or counterexample would be very much appreciated!