Normal subgroup contained in centralizer of derived subgroup

Definition
A subgroup $$H$$ of a group $$G$$ is termed a normal subgroup contained in centralizer of commutator subgroup if $$H$$ is a normal subgroup of $$G$$ and $$H \le C_G([G,G])$$, i.e., $$H$$ is contained in the defining ingredient::centralizer of derived subgroup of $$G$$.

Related group properties
A group $$H$$ has the property that for any group $$G$$ containing $$H$$ as a normal subgroup, $$H$$ is also contained in the centralizer of commutator subgroup of $$G$$, if and only if $$H$$ is a group whose automorphism group is abelian.