Centralizer of derived subgroup has class at most two

Statement

Suppose $G$ is a group and $H=C_{G}([G,G])$ is the Centralizer of commutator subgroup (?) of $G$. Then, $H$ is a Group of nilpotency class two (?). In other words, $H$ is either an abelian group or a non-abelian group of class two. In particular, $H$ is a Class two normal subgroup (?) and a Class two characteristic subgroup (?).

Related facts

• Commutator subgroup centralizes cyclic normal subgroup
• Commutator subgroup centralizes aut-abelian normal subgroup
• Centralizer of commutator subgroup is hereditarily 2-subnormal
• Abelian Frattini subgroup implies centralizer is critical

Proof

Given: A group $G$. $H=C_{G}([G,G])$.

To prove: $H$ is a group of class (at most) two.

Proof: Since $[G,G]$ centralizes $H$, and $H\leq G$, $[H,H]$ centralizes $H$. Thus, $[H,H]$ is contained in the center of $H$. Thus, $H$ has class at most two.