Let $F\_2$ be the free group of rank 2. Let $K\le F\_2$ be a characteristic subgroup, such that $G := F\_2/K$ is finite.


Do there exist examples of such **nonabelian** $G$ such that the induced map
$$Aut(F\_2)\rightarrow Aut(G)$$
is surjective?


For example, if $C\_n$ is the cyclic group of order $n$, then for every $n$, $G = C\_n\times C\_n$ is an abelian example of such a group.


EDIT: As Derek Holt pointed out in the comments, $C\_n\times C\_n$ is only an example in the cases $n = 1,2,3,4,6$.



are there finite nonabelian characteristic quotients $G$ of $F_2$ inducing a surjection $Aut(F_2)\twoheadrightarrow Aut(G)$?
I found one example of a verbal subgroup (as suggested by Arturo Magidin) that worked, but others that did not. Let $F/K$ be the largest quotient of $F=F\_2$ that is a $2$-group and has exponent $2$ class $2$. In other words, $K=H^2[F,H]$, where $H=F^2[F,F]$.


Then $|F/K| = 2^5$ and has automorphism group of order $384$. (That is made of ${\rm GL}(2,3)$ of order $6$ acting on $F/H$, with a group of order $2^6$ that centralizes the layers $F/H$ and $H/K$.)


Now ${\rm Aut}(F)$ with $F = \langle a,b \rangle$ is generated by the three automorphisms (i) $a \mapsto b$, $b \mapsto a$; (ii) $a \mapsto ab$, $b \mapsto b$; and (iii) $a \mapsto a^{-1}$, $b \mapsto b$. It can be checked (I checked on a computer) that the induced automorphisms of $F/K$ generate the whole of ${\rm Aut}(F/K)$.


However, when I tried the same computation with exponent $3$ in place of exponent $2$, I found that $|{\rm Aut}(F/K)| = 34992 = |{\rm GL}(2,3)| \times 3^6$, but the subgroup induced by ${\rm Aut}(F)$ has order only $11664$.


I also tried the exponent $2$ class $3$ quotient of $F/L$ of $F$, where $L=K^2[F,K]$. Then $|F/L|=2^{10}$ and has automorphism group of order $6 \times 2^{16}$, but the subgroup induced by ${\rm Aut}(F)$ has order only $6 \times 2^{12}$.


This suggests that there might not be an abundance of examples.


*Added later*: Luc Guyot has pointed out that there is a quotient of $G/K$isomorphic to $Q\_8$ which is also an example. I found also that the intersection of the kernels of the homomorphisms from $F$ to $S\_3$, which has index $108$ in $F$, is a further example.


