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I will refer to a group as self automorphic if it is isomorphic to its automorphism group, that is, its group of automorphisms. For example, the group $S_3$ is self automorphic.

I have two questions:
(1) Is there a classification of the finite self automorphic groups? If so please let me know where I can read about this classification.
(2) Is there a classification of the countable self automorphic groups?

Seeing as that there apears to not be a complete classification of these groups, could you give an example other than a symmetric group or the group $D_8$? Thanks

Mathew
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    I don't believe a full list is known. See this for more information. – lulu Jan 03 '20 at 21:56
  • are you saying you beleive a full list is not known for the finite case or the countable case? Thanks – Mathew Jan 03 '20 at 21:58
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    A believe that no full list is known in either case. To be clear: I could be wrong about that. – lulu Jan 03 '20 at 21:58
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    here is more information on the finite case. – lulu Jan 03 '20 at 21:59
  • No, a classification sounds hopeless even in the realm of finite groups. Still, many subcases are known, e.g., among finite Coxeter groups, or among finite simple groups. – YCor Jan 06 '20 at 17:40

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