  A full characterization of when a subgroup $H$ of a group $G$ in a varietal
product ${\cal NQ}$ is epimorphically embedded in $G$ (in the variety ${\cal
NQ}$) is given. From this, a result of S.~McKay is derived, which states that
if ${\cal NQ}$ has instances of nonsurjective epimorphisms, then ${\cal N}$
also has instances of nonsurjective epimorphisms. Two partial converses to
McKay's result are also given: when~$G$ is a finite nonabelian simple group;
and when~$G$ is finite and ${\cal Q}$ is a product of varieties of nilpotent
groups, each of which contains the infinite cyclic group.
