My motivation for this question is from Universal Algebra: A congruence of an arbitrary algebra $A$ is said to be principal, if it is generated by a single element. In the case of rings, this is just the notion of principal ideal and for groups it is a normal subgroup which is the normal closure of a single element, more precisely:

A normal subgroup of the form $\langle x^G\rangle$ is called a principal subgroup of the group $G$. We say that $G$ is a principal group, if every normal subgroup of $G$ is principal.

Is there any classification of principal groups? Is there at least a classification of nilpotent (solvable) principal groups?

The same notion can be defined for Lie algebras and also the same questions for Lie algebras arise.