  A finitely generated group $\Gamma$ is called strongly scale-invariant if
there exists an injective endomorphism $\varphi: \Gamma \to \Gamma$ with the
image $\varphi(\Gamma)$ of finite index in $\Gamma$ and the subgroup
$\displaystyle \bigcap_{n>0}\varphi^n(\Gamma)$ finite. The only known examples
of such groups are virtually nilpotent, or equivalently, all examples have
polynomial growth. A question by Nekrashevych and Pete asks whether these
groups are the only possibilities for such endomorphisms, motivated by the
positive answer due to Gromov in the special case of expanding group morphisms.
  In this paper, we study this question for the class of virtually polycyclic
groups, i.e. the virtually solvable groups for which every subgroup is finitely
generated. Using the $\mathbb{Q}$-algebraic hull, which allows us to extend the
injective endomorphisms of certain virtually polycyclic groups to a linear
algebraic group, we show that the existence of such an endomorphism implies
that the group is virtually nilpotent. Moreover, we fully characterize which
virtually nilpotent groups have a morphism satisfying the condition above,
related to the existence of a positive grading on the corresponding radicable
nilpotent group. As another application of the methods, we generalize a result
of Fel'shtyn and Lee about which maps on infra-solvmanifolds can have finite
Reidemeister number for all iterates.
