Being characteristic is transitive

So we have $\,H\,$ char $\,N\,$ char $\,G\,$ . Let

$$\phi\in \operatorname {Aut}(G)\Longrightarrow \phi(N)=N\,\,,\,\text{since}\,\,N\,\,\text{is characteristic in}\,\,G$$

but this means $\,\left.\phi\right|_N\in\operatorname{Aut}(N)\,$ , so

$$\phi(H)=\left.\phi\right|_N(H)\stackrel{\text{since}\, H\,\mathbf{char}\, N}=H\Longrightarrow H\,\,\mathbf {char}\,G$$