Let $X=\{0,1\}^{\mathbb{N}}$ and $\theta$ be the partition of $X$ induced by the equivalence relation $x \sim x'$ when $x$ and $x'$ differ only at a finite number of coordinates (see this related question).

Given a Bernoulli measure $m$ on $X$, let ${\cal H}$ be the group of transformations $S$ of $X$ satisfying $\theta(x)=\theta(S(x))$ for almost all $x$ and let ${\cal G}$ be the subgroup of ${\cal H}$ consisting of measure-preserving transformations.

Is it possible to explicitely describe ${\cal G}$ ? Under which condition on $m$ the group ${\cal G}$ is ergodic ?

EDIT: I am also interested in the case when $m$ is a stationnary Markov probability on $X$.