Given two classes $ {\mathfrak{F}} $ and $ {\mathfrak{X}} $ of finite groups, $ {\mathfrak{F}} $ is said to have property $ {\mathcal{P}}_{2} $ for $ {\mathfrak{X}} $ whenever $ {\mathfrak{F}} $ contains every $ {\mathfrak{X}} $ -group $ G $ expressible as the product of some subgroups $ A_{1},A_{2},\dots,A_{n} $ such thatthe groups $ A_{i}A_{j} $ lie in $ {\mathfrak{F}} $ for all $ 1\leq i<j\leq n $ .This article describes all $ Z $ -saturated $ s_{F} $ -closed formationsand Fischer formations of solvable groupswith property $ {\mathcal{P}}_{2} $ .In particular,the set of all such formations coincides withthe set of hereditary Shemetkov formationsin the class $ {\mathfrak{S}} $ of all finite solvable groups.We describe the hereditary saturated formations $ {\mathfrak{X}} $ with every saturated subformation having property $ {\mathcal{P}}_{2} $ for $ {\mathfrak{X}} $ .
