Let \(G\) be a group. A subgroup \(H\leq G\) is characteristic in \(G\), written \(H \ch G\), if \(\phi(H) \leq H\) for every \(\phi \in \Aut(G)\).
If \(H \ch G\), then \(\phi(H) = H\) for every \(\phi\in\Aut(G)\): applying the definition to \(\phi^{-1}\) gives \(\phi^{-1}(H)\leq H\), so \(H\leq\phi(H)\). The automorphisms need not fix \(H\) pointwise.