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Given a centerless finite group G, with at least one automorphism which is not conjugation by an element of G. Is there any lower bound on the size of Aut(G) given in terms of G? (As big as possible, of course).

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Only the trivial bound $2|G|$, because the alternating groups $A_n$, $n\neq 6$ make this sharp. See e.g. the Wikipedia article.

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