Let $\Gamma$ be a countable group and let $\Lambda_1,\Lambda_2<\Gamma$ be subgroups. We say that $\Lambda_1$ is amenable relative to $\Lambda_2$ if the action of $\Lambda_1$ on $\Gamma/\Lambda_2$ by left translation admits an invariant mean.
Let now $\Sigma<\Lambda<\Gamma$ be subgroups and let $G<N_\Gamma(\Sigma)$ be a subgroup of the normalizer of $\Sigma$ in $\Gamma$. Assume that $G$ is amenable relative to $\Lambda$.
Question: Is $\Sigma G$ still amenable relative to $\Lambda$ ?