Examples of polycyclic groups with regular automorphisms of order 4

We construct torsion-free polycyclic groups with fixed-point-free automorphisms of order 4 that are not nilpotent-by-finite, not metabelian-by-finite, not nilpotent-by-abelian and hence not centre-by-metabelian. (Note that such groups are always finite extensions of centre-by-metabelian groups.)