Suppose we have a group $G$ with subgroups $H,H'$ and an isomorphism $\phi:H\to H'$. My question is when it is possible to extend $\phi$ to an automorphism on the whole group $G$? A slightly weaker version of this question would be to ask whether we can find any automorphism of $G$ which restricts to an isomorphism between $H$ and $H'$. Phrased differntly, for which groups is the answer to the above question yes for any choice of isomorphic subgroups?

A related question was asked in this post but no answer was provided there about which groups do have this lifting property.

Clearly this property holds for cyclic groups. In the example of the linked post they show that it does not hold for $S_4$, and a similar argument shows that it doesn't hold for $S_n, n\geq 4,n\neq 6$. Another example where this property fails is in the group $G := \mathbb{Z}/2 \times \mathbb{Z}/4$ as there is not automorphism which maps $\langle(1,0)\rangle$ to $\langle (0,2)\rangle$. I would really love a classification of the groups where this property holds, or at least a start towards that.

Edit: Thank you everyone for your helpful contributions. Since someone asked, here is a more precise characterization of what I'm looking for.

I'm most interested in groups $G$ which have the property that for any pair of isomorphic subgroups $H,H' \leq G$, there exists an automorphism $\rho$ of $G$ which restricts to an isomorphism $\rho\mid_H:H\to H'$.

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    $\begingroup$ I wish you'd edit this for clarity. You have a question, then another question then something which is apparently a different way of asking one or other of these questions (although as now $H$ varies I can't see that), then later you speak of "this property" .... $\endgroup$ Jul 15, 2022 at 9:25
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    $\begingroup$ It doesn't hold for $S_n$, $n\geq 4$, (including $n=6$) because $(1,2)$ can never be sent to $(1,2)(3,4)$, as one is outside $A_n$ and one inside $A_n$. For abelian $p$-groups it holds only for direct products of isomorphic cyclic $p$-groups, e.g., $C_4\times C_4$. In general it almost never holds for a group. $\endgroup$ Jul 15, 2022 at 9:51
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    $\begingroup$ A more general class is one where any two cyclic subgroups of the same order are conjugate. (Although this is just conjugate, rather than via automorphism.) This has been looked at before, e.g., tandfonline.com/doi/abs/10.1080/00927870902828835 In the Kourovka notebook, 7.48, it's stated that if any two elements of the same order in $G$ are conjugate in $G$, then $|G|\leq 2$, which is related to your question. $\endgroup$ Jul 15, 2022 at 9:55
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    $\begingroup$ There is a particular case that is interesting in its own sake. Let $G$ be the symmetric group on some non-empty set. Let $H, K$ be two regular subgroups of $G$. If $f : H \to K$ is an isomorphism, and then there is $g \in G$ such that $f(H) = gˆ\{-1\} h g$ for $h \in H$. In other words, $f$ extends to an (inner) automorphism of $G$. $\endgroup$ Jul 15, 2022 at 10:43
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    $\begingroup$ Examples of non-abelian $2$-groups with the preoprty include Suzuki $2$-groups, where all elements of order $2$ are central and there is an automorphism cyclically permuting them. $\endgroup$ Jul 15, 2022 at 13:00

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For groups of odd order with your property for groups of order $p$ only, so a larger class of groups, this paper from 1973 gives a structural result. They are pretty close to direct products of $p$-groups: they are a direct product $H$ of $p$-groups for various primes $p$, each of which has this property, and then a cyclic group of automorphisms of order prime to $|H|$ can act on this.

I don't think the subject has received much attention apart from this result, although in the comments I mentioned a paper on $p$-groups with this property. Using CFSG, which was unavailable in 1973, it should be possible to prove a version of this result for groups of even order.

Edit: so do $A_5=PSL_2(5)$ and $PSL_2(7)$. I haven't checked but it might be true for $PSL_2(p)$.


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