Coamenability and strong ergodicity

Following methods of Bannon-Marrakchi-Ozawa, we show that for coamenable inclusion $\mathcal{S}\le \mathcal{R}$ of ergodic, probability measure-preserving relations, we have that $\mathcal{R}$ is strongly ergodic if and only if $\mathcal{S}$ is strongly ergodic. More general results are given when $\mathcal{S}\le \mathcal{R}$ is coamenable, $\mathcal{R}$ is strongly ergodic, but we do not assume ergodicity of $\mathcal{S}$. As a consequence, if $\mathrm{\Lambda }\le \mathrm{\Gamma }$ is a coamenable inclusion of groups, then any strongly ergodic $\mathrm{\Gamma }$ action has countably many ergodic components for the $\mathrm{\Lambda }$ action, each of which is strongly ergodic.