We provide a polynomial time algorithm for deciding the equation solvability problem
over finite groups that are semidirect products of a p-group and an Abelian group.
As a consequence, we obtain a polynomial time algorithm for deciding the equivalence
problem over semidirect products of a finite nilpotent group and a finite Abelian
group.
The key ingredient of the proof is to represent group expressions using a special
polycyclic
presentation of these finite solvable groups.
