Centralizer of derived subgroup is hereditarily 2-subnormal

This article gives the statement, and possibly proof, of the fact that for any group, the subgroup obtained by applying a given subgroup-defining function (i.e., centralizer of commutator subgroup) always satisfies a particular subgroup property (i.e., hereditarily 2-subnormal subgroup)}
View subgroup property satisfactions for subgroup-defining functions $|$ View subgroup property dissatisfactions for subgroup-defining functions

Statement

Suppose $G$ is a group and $H=C_{G}([G,G])$ is its centralizer of commutator subgroup. In other words, $H$ is the centralizer of the commutator subgroup of $G$. Then, $H$ is a hereditarily 2-subnormal subgroup of $G$: every subgroup of $H$ is a 2-subnormal subgroup (?) of $G$.

Related facts

• Centralizer of commutator subgroup has class at most two
• Abelian Frattini subgroup implies centralizer is critical
• Commutator subgroup centralizes cyclic normal subgroup
• Commutator subgroup centralizes aut-abelian normal subgroup