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 $G = \mathbb{Z}^2$.