Let G be a finite group and consider a field K of prime characteristic p. Let P be the projective cover of the trivial KG-module, which we denote by K, and J the Jacobson radical J(KG) of the group algebra KG. Let e 2 KG be a primitive idempotent such that P = eKG. We are concerned with the second term eJ=eJ 2 of the lower Loewy series of P . It is a completely reducible KG-module, whose composition factors are just the irreducible KG-modules V such that there exists a nonsplit KG-module extension 0 ! V ! E ! K ! 0 (see [7, VII 16.8]). Gaschutz (see [7, VII §15]) gives a complete description of eJ=eJ 2 for K = Fp, the field of p elements, and G a p-soluble group: Its composition factors are precisely the abelian complemented p-chief factors of G, counting the multiplicities. Later Willems shows [12] that for any G each complemented p-chief factor of G appears as a component of eJ=eJ 2 with multiplicity not less than that as a (complemented) chief factor of G. Okuyama and Tsushima [10] define a filtration of eJ=eJ 2 from a chief series of G, which provides a new proof of these results and makes explicit the relationship between the chief factors of G and the composition factors of eJ=eJ 2. In this paper we give a description of eJ=eJ 2 for any G and any field K of characteristic p, which only depends on the knowledge of what occurs for certain almost simple sections of G, by means of the development of a reduction theorem of Kovacs [8]. As an application we obtain the terms of the filtration of Okuyama and Tsushima corresponding to any chief factor of any G.