  The study of the complexity of the equation satisfiability problem in finite
groups had been initiated by Goldmann and Russell (2002) where they showed that
this problem is in polynomial time for nilpotent groups while it is NP-complete
for non-solvable groups. Since then, several results have appeared showing that
the problem can be solved in polynomial time in certain solvable groups $G$
having a nilpotent normal subgroup $H$ with nilpotent factor $G/H$. This paper
shows that such normal subgroup must exist in each finite group with equation
satisfiability solvable in polynomial time, unless the Exponential Time
Hypothesis fails.
