# On the existence of a group Automorphism of order 2

Is it true that every finite group of even order $>2$ has an automorphism of order $2?$

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This is not true. For each odd prime $p,$ there are finite $p$-groups whose automorphism group is a $p$-group (in fact, "most" $p$-groups have that property in an appropriate sense- see work of U. Martin and coauthors). Take the direct product of such a $p$-group with a cyclic group of order $2$ and the automorphism group of the resulting group is still a $p$-group.
To find exampless of $p$-groups whose automorphism group is a $p$-group is not hard. I think there are example in Gorenstein's text "Finite Groups" (Harper and Row, 1968). This is all you really need. The fact that almost all $p$-groups have automorphism group a $p$-group is what Ursula Martin and coathors proved, but that is not essential to answer your question, it was for general interest. –  Geoff Robinson Dec 25 '11 at 18:56