Is the commutator subgroup of the commutator subgroup still a normal subgroup?

Question

Suppose that $G$ is any group, let $G_1:=[G,G]$ and $G_2:=[G_1,G_1]$. Is it true that $G_2$ is still a normal subgroup of $G$ or are there exceptions? I've tried coming up with a homomorphism with kernel $G_2$, but I couldn't really get anything started. Does anyone know if this is true or not?

Answer 1

Yes, because the commutator subgroup is characteristic, see here:

Commutator subgroup $G'$ is a characteristic subgroup of $G$

If $N$ is a normal subgroup of a group $G$, and $M$ is a characteristic subgroup of $N$, then $M$ is a normal subgroup of G.