On some questions of Eymard and Bekka concerning amenability of homogeneous spaces and induced representations

Let $F\subseteq H\subseteq G$ be closed subgroups of a locally compact group. In response to a 1972 question by Eymard, we construct an example where the homogeneous factor-space $G\mathrm{/}F$ is amenable in the sense of Eymard-Greenleaf, while $H\mathrm{/}F$ is not. (In our example, $G$ is discrete.) As a corollary which answers a 1990 question by Bekka, the induced representation ${\text{ind}}_H^G(\rho )$ can be amenable in the sense of Bekka even if $\rho$ is not amenable. The second example, answering another question by Bekka, shows that ${\text{ind}}_H^G(\rho )$ need not be amenable even if both the representation $\rho$ and the coset space $G\mathrm{/}H$ are amenable.