The answer seems to be affirmative. We use the idea of Henry Wilton that the image might be taken as an alternating group $A_q$, a <i>simple</i> one (see his comment above). Let $K=\mathcal A(1).$ Then

$\mathcal A(m) \ge [K,K,\ldots,K]=[..[K,K],..,K]\qquad (m\quad times) \qquad (*)$

Take a nontrivial $\alpha \in [K,K]$ and a surjective homomorphism $\Delta: \mathrm{Aut}(F_n) \to A_q$ which doesn't vanish at $\alpha$.

Then 
$$  
A_q =\mathrm{NormalClosure}(\Delta(\alpha))=\Delta([K,K])=\Delta(K).  
$$  
It follows that 
$$  
\Delta( [K,K,\ldots,K])=A_q  
$$ 
and by $(*)$  $\Delta( \mathcal A(m))=A_q  $ for every $m \ge 1.$