Ergodic decomposition of the action of a subgroup

Question

Let $G$ be a countable abelian group and let $H \le G$ be a subgroup. Let $G \curvearrowright (X,\mu)$ be an ergodic measure preserving action on some probability space $(X,\mu)$. Now we know that the action $H \curvearrowright (X,\mu)$ may not be ergodic, but it has an ergodic decomposition $\mu = \int_Y\mu_y d\nu_y$ where each $\mu_y$ is ergodic for the action of $H$.

Questions:

1. Are almost all the ergodic components $\mu_y$ isomorphic to each other (as $H$-systems)?
2. If $G$ is weakly mixing then are almost all $\mu_y$ weakly mixing?

The answer seems to be yes if $H$ is finite index in $G$. I am interested in the case of $H = \mathbb{Z} \times \{0\} $ and $G = \mathbb{Z}^2$.

Answer 1

The answer to both questions is "no" - a counterexample (labeled "folklore") appears immediately prior to Question 6.6 in Austin, Tim, Extensions of probability-preserving systems by measurably-varying homogeneous spaces and applications, Fundam. Math. 210, No. 2, 133-206 (2010). ZBL1206.28023..

Here it is, essentially verbatim:

Let $G = \mathbb Z\times \mathbb Z$, let $\mathbb T = \mathbb R/\mathbb Z$ with Haar probability measure $m$, and let $X = \mathbb T^2 \times \mathbb T^2$ and $\mu=$ Haar probability measure on $X$. Let $A:\mathbb T^2\to \mathbb T^2$ be any ergodic toral automorphism, and define an action of $\mathbb Z\times \mathbb Z$ on $X$ by $(n,m)\cdot (\mathbf x, \mathbf y) = (A^n\mathbf x, A^n(\mathbf y+m\mathbf x))$. In other words, $G \curvearrowright (X,\mu)$ is the action generated by $A\times A$ and the map $(\mathbf x,\mathbf y)\mapsto (\mathbf x, \mathbf y+\mathbf x)$. The action generated by $A\times A$ is ergodic (and in fact mixing), since the action generated by $A$ is mixing. The entire action is therefore weak mixing.

Taking $H:= \{(0,m):m\in \mathbb Z\}$, the ergodic components are given by rotation $(0,m)\cdot \mathbf y = \mathbf y + m \mathbf x$ on the vertical $Y_{\mathbf x}:=\{(\mathbf x, \mathbf y): \mathbf y\in \mathbb T^2\}$. Examining the spectrum of these components reveals that they are not (almost all) mutually isomorphic, and no component is weak mixing.