Let $F_n$ where $n \ge 3$ be a free group and let $(\mathcal A_n(k))$ where $k \ge 1$ be the kernel of the homomorphism $Aut(F_n) \to Aut(F_n/\gamma_{k+1}(F))$ determined by the natural homomorphism $F_n \to F_n/\gamma_{k+1}(F).$

($(\mathcal A_n(k) : k \ge 1)$ is called the Johnson filtration of $Aut(F_n);$

$\gamma_k(G)$ denotes the $k$-th terms of the lower central series of a group $G,$ $\gamma_1(G)$ being equal to $G$).

I do not know an example of a group homomorphism
$f : Aut(F_n) \to G$ which
takes * all* terms of the Johnson filtration
$(\mathcal A_n(k))$ to the same

*subgroup: $$ 1 \ne f(\mathcal A_n(1))=f(\mathcal A_n(2)) = \ldots = f(\mathcal A_n(k)) = \ldots $$ I would be very grateful for such an example, or for an argument that homomorphisms like that do not exist.*

**nontrivial**
Canad. J. Math.29 (1977), no. 3, 541--551. – HJRW Aug 12 '10 at 21:31