Normal closure of a subgroup of a free group.

Question

Let $G$ be a finitely generated free (nonabelian) group, $H$ a subgroup generated by some of the generators of $G$, and $a: G\to AG$ be the projection to the abelianization $AG:=G/[G,G]$.

Is it true that the normal closure $N(H)$ equal $a^{-1}(H)$?

In other words, can I rewrite every element in $H \cdot [G,G]$ as a product of elements of type $ghg^{-1}$?

Thanks for possible hint!

(Edit: I'm mostly interested in the case when $H$ is nontrivial, thx @Tsemo Aristide)

Answer 1

$N(H)$ normally denotes the normalizer of $H$, not its normal closure! I would denote its normal closure by $\langle H^G \rangle$.

The answer is usually no, because $H/\langle H^G \rangle$ is isomorphic to the free group on the generators of $G$ that are not in $H$, which is nonabelian provided that there are at least two such generators, but $a^{-1}(H)$ contains $[G,G]$, so $G/a^{-1}(H)$ is abelian.

Answer 2

It is not true, suppose $H=\{e\}$ the neutral element. $a^{-1}(1)=[G,G]$, but $H$ is normal, thus $H$ is equal to its normal closure.