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By definition, if $H$ is characteristic subgroup of $G$, then every automorphism of $G$ restricted on $H$ is an automorphism of $H$.

From these two questions (1) (2) we know it's generally false that if $H$ is characteristic subgroup of $G$, given $\varphi \in \text{Aut}(H)$, then there exists $\psi\in\text{Aut}(G)$ s.t. $\psi|_H=\varphi$. Example from $A_3$ and $S_3$ shows even if such a $\psi$ exists, it may not be unique.

My question:

Suppose $H$ is subgroup of $G$, what's the sufficient condition s.t. $\forall \varphi \in \text{Aut}(H)$, $\exists\ \psi\in\text{Aut}(G)$ s.t. $\psi|_H=\varphi$?

Background:

This question comes from studying $\text{Aut}(S_n)$ and $\text{Aut}(A_n)$.

If $n\geqslant 4$ and $n \neq 6$, then $\operatorname{Aut}(A_n)\cong\operatorname{Inn }(S_n)\cong\operatorname{Aut}(S_n)\cong S_n$.

The existence and uniqueness is true for $A_n$ and $S_n$($n\geqslant 4$ and $n \neq 6$), while uniqueness is not true for $A_3$ and $S_3$.

To move this question out of the unanswered list, I put related materials in answer.

Andrews
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    See also this question. https://math.stackexchange.com/questions/1477524/characteristic-subgroups-and-automorphisms – Levi Oct 31 '19 at 12:19
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    This seems extremely broad. What kind of answer are you expecting? – user1729 Oct 31 '19 at 16:52
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    It's an inverse problem for characteristic subgroup, and now that it's doesn't hold for $H$ just being characteristic subgroup of $G$, I wonder if there should be some restriction on subgroup $H$ of $G$, maybe index of $H$ in $G$ and center of $H$(like the $S_n$ and $A_n$ case), commutability of $H$, existing group $K$ s.t. $G \cong H\rtimes K$ etc. – Andrews Nov 01 '19 at 05:26