It is well known that a finite group admitting an automorphism of order 2 that fixes only the identity is abelian and has odd order. Moreover, the automorphism is inversion.


Is anything known about finite groups admitting an automorphism of order 2 that fixes only the identity and one other element?



Groups with an automorphism of order two fixing only two elements
MacKay [On the structure of a special class of $p$-groups, Quart. J. Math. Oxford Ser (2) 38, 489-502] and, indipendently, Kiming [Structure and derived length of finite $p$-groups possessing an automorphism of $p$-power order having exactly $p$ fixed points, Math. Scand. 62, 153-172] showed that if a finite $p$-group $G$ admits an automorphism of order $p^n$ with exactly $p$ fixed points, then $G$ contains a subgroup $H$ of index bounded by a function of $p$ and $n$ which is nilpotent of class at most 2 (and $H$ is abelian if $p=2$).


