This is an improved version of my previous question, where I forgot to put one of the assumptions.

Question. Let $G$ be a finitely generated non-amenable discrete group, and $H$ be a subgroup of $G$ of infinite index, such that no finite-index subgroup of $H$ is normal in $G$. Can it happen that the index of the normalizer $N(H)$ of $H$ in $G$ is 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. It would be especially nice if $G$ could be taken to be a property $(T)$ group.

I hope I got everything right this time, but if the question is very easy then please point out the example in the comments, so that I can improve the question without starting a new thread.

The motivation for the question is that out of such an example one can construct an ergodic, faithful, non-free action of a non-amenable group whose equivalence relation is amenable. (Most likely such examples have been known before.)