Potentially iterated commutator subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a potentially iterated commutator subgroup of $$G$$ if there exists a group $$K$$ containing $$G$$ such that $$H$$ is an defining ingredient::iterated commutator subgroup of $$K$$, i.e., $$H$$ is obtained from $$K$$ through a series of operations that involve taking the commutator of two subgroups. (special cases of this are when $$H$$ is in the finite part of the derived series or the lower central series of $$K$$).