$\mathfrak F$-projectors and $\mathfrak F$-covering subgroups of finite groups

Given a nonempty set $\omega$ of primes and a nonempty class $\mathfrak F$ of groups, we define the $\mathfrak F^\omega$-projector and $\mathfrak F^\omega$-covering subgroup of a finite group and study their properties (existence, invariance under certain homomorphisms, conjugacy, and embedding). We extend the results of Gaschütz, Schunck, Erickson, and others.