All groups considered in this paper are finite.

The origin and further development of Class Theory has as a corner-stone the concepts of saturated formation and covering subgroup. These were first introduced by Gaschütz in 1963 in his paper Zur Theorie der endlichen auflösbaren Gruppen with the aim of building a general context in which the properties of existence and conjugacy of Sylow subgroups, Hall subgroups and Carter subgroups could appear as particular cases. Thus, working in a universe V of groups which is closed with respect to the usual closure operators, a formation is a class of groups F contained in V with the following properties:

1. Every homomorphic image of an F-group is an F-group.

2. If G/M and G/N are F-groups, then G/(M ∩ N) is also an F-group.

The formation F is said to be saturated if the group G belongs to F whenever the Frattini factor group G/Φ(G) is in F.

For a given group G ∈ V and a class of groups F, an F-covering subgroup of G is a subgroup C of G belonging to F and such that whenever C ≤ H ≤ G, K ⊴ H and H/K ∈ F, then H = CK. So C covers each F-quotient of every intermediate group of G. When V is the universe of finite soluble groups and F is a saturated formation of soluble groups, Gaschütz proved that every group G ∈ V has a unique non-empty conjugacy class of F-covering subgroups. Also it was shown by him that when F is the class of Sp, Sπ or N, then the F-covering subgroups are the Sylow, Hall and Carter subgroups respectively.