19.37 (2018)

a) Does there exist an absolute constant $k$ such that for any nilpotent injector $H$ of an arbitrary finite group $G$ there are $k$ conjugates of $H$ the intersection of which is equal to the Fitting subgroup $F(G)$ of $G$?
b) Can one choose $k = 3$ as such a constant? This is true for finite soluble groups (D. S. Passman, Trans. Amer. Math. Soc., 123, no. 1 (1966), 99–111; A. Mann, Proc. Amer. Math. Soc., 53, no. 1 (1975), 262–264).