Geodesic
Finite groups admitting a fixed-point-free 2-automorphism
Matematičeskie zametki, Tome 23 (1978) no. 5, pp. 651-657

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It is proved that if a finite group admits a fixed-point-free automorphism of order $2^n$, then its nilpotent length is at most $n$. It had been proved by Gross [1] that its nilpotent length is at most $2n-2$. 
@article{MZM_1978_23_5_a1,
     author = {E. I. Khukhro},
     title = {Finite groups admitting a fixed-point-free 2-automorphism},
     journal = {Matemati\v{c}eskie zametki},
     pages = {651--657},
     publisher = {mathdoc},
     volume = {23},
     number = {5},
     year = {1978},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1978_23_5_a1/}
}
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E. I. Khukhro. Finite groups admitting a fixed-point-free 2-automorphism. Matematičeskie zametki, Tome 23 (1978) no. 5, pp. 651-657. http://geodesic.mathdoc.fr/item/MZM_1978_23_5_a1/