Commutator of a normal subgroup and a subset

Definition
A subgroup $$H$$ of a group $$G$$ is termed a commutator of a normal subgroup and a subset if there exists a defining ingredient::normal subgroup $$N$$ of $$G$$ and a subset $$S$$ of $$G$$ such that $$H$$ is the commutator $$[N,S]$$, i.e., the subgroup generated by all the defining ingredient::commutators between elements of $$N$$ and elements of $$S$$.