For $n \in\mathbb{N}$ define:

• $X_n=\{x_1,\ldots,x_n\}$,
• $F(X_n)$ the free group on $X_n$,
• $\varphi:F(X_n)\to F(X_{n-1})$ an epimorphism defined by $x_i\stackrel{\varphi}{\mapsto} x_i$ for $1\le i\le n-1$ and $x_n\stackrel{\varphi}{\mapsto} 1$.

Is it true that for every IA automorphism $\alpha\in Aut(F(X_n))$ the map $x_i \mapsto \varphi(\alpha(x_i))$ for $1\le i\le n-1$ defines an element in $Aut(F(X_{n-1}))$?