On a co-induction question of Kechris

This note answers a question of Kechris: if $H<G$ is a normal subgroup of a countable group $G$, $H$ has property MD and $G\mathrm{/}H$ is amenable and residually finite then $G$ also has property MD. Under the same hypothesis we prove that for any action $a$ of $G$, if $b$ is a free action of $G\mathrm{/}H$, and $b_G$ is the induced action of $G$ then ${\text{CInd}}_H^G(a\mathrm{\mid }H)\times b_G$ weakly contains $a$. Moreover, if $H<G$ is any subgroup of a countable group $G$, and the action of $G$ on $G\mathrm{/}H$ is amenable, then ${\text{CInd}}_H^G(a\mathrm{\mid }H)$ weakly contains $a$ whenever $a$ is a Gaussian action.