A subgroup V of G is called an j in jector of G if Vc~ N is F-maximal in N (that is, Vn N is a maximal ~-subgroup of N) for each subnormal subgroup N of G. It is known that the ~-injectors of a group G form a conjugacy class of subgroups 1,2]. If ~-injectors are always normal then ~ is called a normal Fitting class. In I-1] and [-3] some work has been done to classify normal Fitting classes. It has been shown in these papers that to any normal Fitting class there corresponds an object called a Fitting pair (f, A) defined as follows: A is an Abelian group and to each finite soluble group G there exists a homomorphism fG: G~A such that for all N ~ Gf~ = f~lN, where f~lN is the restriction of fG to N. Each such pair defines a Fitting class ~ given by G ~ if and only if fG(G)= 1, 1,Satz 3.1, 1.]. We use this to construct a Fitting class which contains $3, the symmetric group on 3 letters but does not contain the dihedral group of order 18. For some time it had been undecided whether the Fitting class generated by S 3 was the class consisting of all 3-groups extended by 2-groups. For a general survey see the paper of Cossey read at the conference on group theory held in Canberra, Australia 1973. The construction will actually provide uncountably many normal Fitting classes for all of which, the injectors have index 2. We will define and prove the existence of these classes by using Fitting pairs. Let f2 be the set of powers of odd primes and let 0 E f2. Let A be the cyclic group of order 2 and define, for any finite soluble group G,