Polycyclic groups admitting a regular automorphism of order four

Let G be a polycyclic group and . a regular automorphism of order four of G. If the map .: G . G defined by g . = [g,.] is surjective, then the second derived group of G is contained in the centre of G. Abandoning the condition on surjectivity, we prove that C G (. 2) and G/[G, . 2] are both abelian-by-finite.