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.