  A variety of groups does not contain all metabelian groups if and only if
there is an absolute bound for the nilpotency classes of powerful $p$-groups in
the given variety. Similarly, a variety contains only finitely many finite
$p$-groups of any given coclass if and only if not every group that is an
extension of an abelian group by an elementary abelian $p$-group belongs to
that variety.
