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/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/H, and bG is the induced action of G then CInd G H(a|H)×bG weakly contains a. Moreover, if H < G is any subgroup of a countable group G, and the action of G on G/H is amenable, then CIndH(a|H) weakly contains a whenever a is a Gaussian action.