Here, I wanted to verify that:
The property of being characteristic is a transitive relation among subgroups of a group $G$.
For subgroups $N,H\leq G$, we have $N$ char $G$ and $H$ char $N$. So $N$ char $G$ implies: $\forall\psi\in Aut(G); \psi(N)=N$ $H$ char $N$, so for all elements in $Aut(N)$, $H$ is remained never changing. Especially, when I take $\psi'=\psi|_{N}$ then $\psi':N\to Aut(N)$ would be in $Aut(N)$ and $\psi'(H)=H$. Since the last equality is true for $H$ and the maps, caused by restriction on $N$, then I have $H$ char $G$.
Honestly, I am inly not satisfied form the conclusion here and think I am losing something. Thanks for your hints.