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 ‘∀’.