Finite groups with given properties of critical subgroups

We study the structure of the finite groups such that the minimal non-$\mathfrak H$-groups ($\mathfrak H$ is a saturated hereditary formation) are generalized subnormal. In particular, we obtain a detailed description of the structure of the finite groups such that the Schmidt groups are $\mathfrak F$-attainable ($\mathfrak F$ is a saturated hereditary formation with the lattice property).