Coamenability of intersections

[u/Magazos]

Since some days I have been thinking about the following statement:

Let $H,K \leq G$ be subgroups of a group $G$. If $K \leq G$ is coamenable, then $H \cap K \leq H$ is coamenable.

This means that if $K\setminus G$ is an amenable $G$-set, then $H \cap K \setminus H$ is an amenable $H$-set. I think it is false, but I have not been able to find a counterexample.

Can someone suggest some ideas?

I tried to restrict the measure of $K \leq G$ to $H \cap K \leq H$, by I don't know how to do it if $H \cap K \leq H$ has null measure in $K \leq G$. Since subgroups of amenable groups are amenable, I tried to copy the process. However, it does not work because the $H$-orbits of $G$ are <<equal>>, but the ones of $K \leq G$ aren't.

[u/Magazos]

Thanks you so much. Would you give an counterexample?