On the day before yesterday, I read a proof about the transitivity of characteristic subgroups. The proof shouldn’t be too difficult. The key step is just to create a restriction automorphism.
Suppose $K\le H\le G$, and $f\in \Aut(G)$. Then $f(H)=H$. Define the restriction of $f$ on $H$ as $g:H\to H$ such that $g(h)=f(h) \,\forall h \in H$. Clearly, $g\in \Aut(H)$.
I couldn’t understand why $g(K)=K$. I googled “strongly normal transitivity”, but couldn’t find anything relevant. I changed the words, and found the proof on ProofWiki. Then I realised why I got stuck: when I wrote the definition of characteristic subgroup symbolically, I didn’t pay enough attention to ‘∀’.