Groups with nilpotent $n$-generated normal subgroups

Let $L_n({\mathcal N})$ be the class of all groups $G$ in which the normal closure of each $n$-generated subgroup of $G$ belongs to ${\mathcal N}$. It is known that if ${\mathcal N}$ is a quasivariety of groups then so is $L_n({\mathcal N})$. We find the conditions on ${\mathcal N}$ for the sequence $L_1({\mathcal N}),L_2({\mathcal N}),\dots $ to contain infinitely many different quasivarieties. In particular, such are the quasivarieties ${\mathcal N}$ generated by a finitely generated nilpotent nonabelian group.