Product of nilpotent subgroups

We will say that a subgroup X of G satisfies property C in G if C_G(X?X^g)leqq X?X^g{rm C}_{G}(Xcap X^{{g}})leqq Xcap X^{{g}} for all g ? G{g}in G. We obtain that if X is a nilpotent subgroup satisfying property C in G, then XF(G) is nilpotent. As consequence it follows that if Ntriangleleft GNtriangleleft G is nilpotent and X is a nilpotent subgroup of G then C_G(N?X)leqq XC_G(Ncap X)leqq X implies that NX is nilpotent.¶We investigate the relationship between the maximal nilpotent subgroups satisfying property C and the nilpotent injectors in a finite group.