Projectors of finite, solvable groups were defined by Schunk [6]. In this paper, the author defines generalized projectors of finite groups. They are the same as projectors in the case of solvable groups. The existence of this class of subgroups is shown for i7T-solvable groups satisfying certain properties. Let X be a class of groups, which is closed under isomorphisms and 1 & X. DEFINITION. S is x-maximal in G if (i) S is a subgroup of G, (ii) S E X (iii) S(-TszG & Tec XS=T. DEFINITION. S is a X-projector of G if (i) S is a subgroup of G, (ii) H< G-?HS/H is X-max in G/H. DEFINITION. X is called a Schunk class if X satisfies the following conditions: (a) if G E X, H< G then G/H E X, (b) whenever all primitive factor groups of a group G are in X then so is G in X. Schunk [6] proved the following theorem. THEOREM. If X is a Schunk class then every finite, solvable group G has a X-projector and all X-projectors are conjugate. DEFINITION. Let S be a class of subgroups of a finite group G. Then we say S is a characteristic class of conjugate subgroups if (i) H E S and a an automorphism of G then Ha E S, (ii) Ha is a conjugate of H under an element of G i.e., 3g E G such that HHa-H . DEFINITION. A finite group G is called -r,-closed if it contains a normal subgroup N which is a 7T-group and GIN is a 7T'-group. All groups considered in this paper are finite. Let X be any Schunk class consisting of solvable, -r,-closed groups. In this paper we will show that if the 7T'-subgroups of a 7T-solvable group G have X-projectors and they form a characteristic class of conjugate subgroups then G has X-projectors and the X-projectors form a characteristic class of conjugate subgroups. We cannot expect that all X-projectors are conjugate as is clear from the case when G is a finite, nonabelian simple Received by the editors October 18, 1971 and, in revised form, April 26, 1972. AMS (MOS) subject classifications (1969). Primary 2054. ? American Mathematical Society 1973