  A classical result by K.B. Lee states that every group morphism between
almost crystallographic groups is induced by an affine map on the nilpotent Lie
group whereon these groups by definition act. It is the main technique for
studying morphisms between virtually nilpotent groups, having important
applications in fixed point theory, for example as a tool to compute Nielsen
numbers for self-maps on infra-nilmanifolds. In this paper we generalize this
result to morphisms between virtually polycyclic groups, which are known to act
crystallographically by affine transformations on a nilpotent Lie group. The
main method is to study the special class of translation-like actions, which
behave well under taking subgroups and restricting the action to invariant
right cosets.
