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In this paper, we study the structure of polycyclic groups admitting an automorphism of order four on the basis of Neumann’s result, and prove that if $\alpha$ is an automorphism of order four of a polycyclic group $G$ and the map $\varphi :G\to G$ defined by $g^{\varphi }=[g,\alpha ]$ is surjective, then $G$ contains a characteristic subgroup $H$ of finite index such that the second derived subgroup $H^{\text{'}\text{'}}$ is included in the centre of $H$ and $C_H\left({\alpha }^2\right)$ is abelian, both $C_G\left({\alpha }^2\right)$ and $G/[G,{\alpha }^2]$ are abelian-by-finite. These results extend recent and classical results in the literature....