Fixed-point-free automorphism of order three implies nilpotent

Statement
Suppose $$G$$ is a finite group and $$\varphi$$ is an automorphism of $$G$$ of order three. Then, $$G$$ is a nilpotent group and $$g$$ commutes with $$\varphi(g)$$ for every $$g \in G$$.

Related facts

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

Textbook references

 * , Page 336, Theorem 1.5, Section 10.1 (Elementary properties)