  Given a locally compact quantum group $\mathbb{G}$ and a closed quantum
subgroup $\mathbb{H}$, we show that $\mathbb{G}$ is amenable if and only if
$\mathbb{H}$ is amenable and $\mathbb{G}$ acts amenably on the quantum
homogenous space $\mathbb{G}/\mathbb{H}$. We also study the existence of
$L^1(\widehat{\mathbb{G}})$-module projections from
$L^{\infty}(\widehat{\mathbb{G}})$ onto $L^{\infty}(\widehat{\mathbb{H}})$.
