Invariant measures of Toeplitz subshifts on non-amenable groups

Let $G$ be a countable residually finite group (for instance ${\mathbb{F}}_2$) and let $\overleftarrow{G}$ be a totally disconnected metric compactification of $G$ equipped with the action of $G$ by left multiplication. For every $r\ge 1$ we construct a Toeplitz $G$-subshift $(X,\sigma ,G)$, which is an almost one-to-one extension of $\overleftarrow{G}$, having $r$ ergodic measures ${\nu }_1,\cdots ,{\nu }_r$ such that for every $1\le i\le r$ the measure-theoretic dynamical system $(X,\sigma ,G,{\nu }_i)$ is isomorphic to $\overleftarrow{G}$ endowed with the Haar measure. The construction we propose is general (for amenable and non-amenable residually finite groups), however, we point out the differences and obstructions that could appear when the acting group is not amenable.