If $A$ is a subgroup of $B$, just it hold that $\text{Aut}(A)$ is a subgroup of $\text{Aut}(B)$?

Question

Let $\text{Aut}(G)$ denote the group of automorphisms of $G$ and let $A\subseteq B$ denote $A$ is a subgroup of $B$. Does the following hold:

$$A\subseteq B\implies \text{Aut}(A)\subseteq \text{Aut}(B)$$

If not, is there a necessary and sufficient condition for this to hold?

Answer 1

It is certainly not true in general.

For a counterexample take $A:=S_6$ whose automorphism group is well-known to have order $2\cdot 6!$; and take $B:=S_7$ whose automorphism group is well-known to have order $7!=7\cdot 6!$.

Answer 2

This question is hard, and there's not a simple answer. In general, there's no reason a function preserving only the $A$ structure should extend to one preserving all the extra $B$ structure. You can find a discussion about which automorphisms can be extended here, but the answer may not satisfy you.

So I suppose the answer is "no", and the answer to the follow-up question about a necessary and sufficient condition is "not a simple one".

There is also some good discussion in the comments of this question

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I hope this helps ^_^