  It is proved that every group of the form $G=AB$ with two subgroups $A$ and
$B$ each of which is either abelian or has a quasicyclic subgroup of index $2$
is soluble of derived length at most $3$. In particular, if $A$ is abelian and
$B$ is a locally quaternion group, this gives a positive answer to Question
18.95 of "Kourovka notebook" posed by A.I.Sozutov.
