Question 1. Let $G$ be a finitely generated non-amenable discrete group, and $H$ be a subgroup of $G$ of infinite index. Can it happen that the index of the normalizer $N(H)$ of $H$ in $G$ is non-zero finite, and the Schreier graph of $G/H$ has subexponential growth?

If the answer is yes, I would very much like to see an example.