Fixed-point-free automorphism of order four implies solvable

Statement
Suppose $$G$$ is a finite group and $$\varphi$$ is a fixed-point-free automorphism of $$G$$ of order four. Then $$G$$ is a solvable group.

Related facts

 * Fixed-point-free involution on finite group is inverse map
 * Fixed-point-free automorphism of order three implies nilpotent
 * Fixed-point-free automorphism of prime order implies nilpotent

Textbook references

 * , Page 342, Theorem 4.2, Section 10.4 (fixed-point-free automorphisms of order 4)