Let $\mathbb{F}_2$ be the free group of rank $2$ with generators $a$ and $b$. I would like to build an automorphism $\varphi$ of $\mathbb{F}_2$ such that :
1) $\varphi([a,b]) = [a,b]$
2) $\overline{\varphi}$ the induced application on the abelianization of $\mathbb{F}_2 \simeq \mathbb{Z}^2$ is the identity.
For any $n \in \mathbb{Z}$ the conjugation by $[a,b]^n$ does the trick. I was wondering if any such $\varphi$ must be of this form.
(This question is related to the study of elements of the Torrelli group of a closed surface which acts only on a embeded $1$-holed torus).
The main reference is "Primer on mapping class group" by Farb and Margalit. The point is that for once punctured torus the abelianization of the fundamental group defined an isomorphism of the mapping class group to $GL(2,Z)$. Thus, under your assumption the automorphism has to be inner, conjugation by some element $g$ of the free group. Now, use the assumption that it sends $[a,b]$ to itself. This means that $g$ centralizes the commutator. Lastly, the commutator is represented by a simple loop, thus, it generates a maximal cyclic subgroup.