Gradewise properties of subgroups of finite groups

Given a subgroup A of a group G and some group-theoretic property θ of subgroups, say that A enjoys the gradewise property θ in G whenever G has a normal series $$1 = G_0 \leqslant G_1 \leqslant \cdots \leqslant G_t = G$$ such that for each i = 1, …, t the subgroup (A ∩ G i )G i−1/G i−1 enjoys the property θ in G/G i−1. Basing on this concept, we obtain a new characterization of finite supersolvable and solvable groups.