  We characterize coset spaces of topological groups which are coset spaces of
(separable) metrizable groups and complete metrizable (Polish) groups. Besides,
it is shown that for a $G$-space $X$ with a $d$-open action there is a
topological group $H$ of weight and cardinality less than or equal to the
weight of $X$ such that $H$ admits a $d$-open action on $X$. This is further
applied to show that if $X$ is a separable metrizable coset space then $X$ has
a Polish extension which is a coset space of a Polish group.
