Homomorphisms of $\mathbb{F}_2$ that preserve $aba^{-1}b^{-1}$ Let $\mathbb{F}_2$ be the free group generated by $a$ and $b$.  Suppose we are given a homomorphism $\phi: \mathbb{F}_2 \to \mathbb{F}_2$ with the property that $\phi(aba^{-1}b^{-1}) = aba^{-1}b^{-1}$.  Can I conclude that $\phi$ is surjective?  Can I conclude $\phi$ is an isomorphism?   

It's a theorem of Nielsen known as "Nielsen commutator test" (Nielsen, J. Die Isomorphismen der aligemeinen unendlichen Gruppe mit zwei Erzeugenden. Math. Ann. 78 (1918), 385–397.) stating that automorpisms of $F$ = $\langle x, y \rangle$ are precisely those endomorphisms which take $[x, y]$ to any conjugate or inverse to conjugate — it's an easy consequense of the fact that any IA-automorphism of $F$ is inner; if you want, I can write proof here.
It's quite interesting that this result extends to some other 2-generated groups — for example, free 2-generated metabelian group (by V. Durnev) and  "most" groups of type $F/[[R, R], F]$ (by N. Gupta and V. Shpilrain) also satisfy commutator test.
