$\newcommand{\mc}{\mathcal}$ $\newcommand{\Z}{\mathbb Z}$ Let $(X, \mc X, \mu)$ be a probability space and $G$ be a countable group. Let $T:G\times X\to X$ be a measure preserving action of $G$ on $X$. Let $\nu$ be a probability measure on $G$ such that the support of $\nu$ contains a generating set of $G$. Equip $G^\Z$ with the probability measure $\nu^\Z$ and let $\sigma:G^\Z\to G^\Z$ be the shift map. We will write an element of $G^\Z$ as $g = (g_n)_{n\in \Z}$.
We define a map $\Phi:X\times G^\Z\to X\times G^\Z$ as $$ \Phi(x, g) = (T^{g_0}x, \sigma(g)) $$ By Fubini's theorem it follows that $\Phi$ preserves the measure $\mu\times \nu^{\Z}$.
Let $\mc I$ be the $\sigma$-algebra on $X$ consisting of all the $G$-invariant subsets of $X$.
Question. Can we describe the $\sigma$-algebra of $\Phi$-invariant subsets of $X\times G^{\mathbb Z}$ in terms of $\mc I$.
It is clear that for any $A\in \mc I$ we have $A\times G^\Z$ is $\Phi$-invariant. Perhaps these are all the $\Phi$-invariant subsets. The question is important since in order to get interesting information by applying ergodic theorems to the map $\Phi$ it would be useful to know the $\Phi$-invariant $\sigma$-algebra.