Let $F_2$ be the free group of rank 2. What is the abelianization of $\operatorname{Aut}(F_2)$?
There is a surjection $\operatorname{Aut}(F_2) \rightarrow\operatorname{ GL}_2(\mathbb{Z})$, so we at least have $\mathbb{Z}/2$ as a quotient. Is this the full abelianization?