On $frak {F}$-normal Fitting classes of finite soluble groups

Let frak X, frak F,frak Xsubseteqq frak Ffrak {X}, frak {F},frak {X}subseteqq frak {F}, be non-trivial Fitting classes of finite soluble groups such that G_{frak X}G_{frak {X}} is an frak Xfrak {X}-injector of G for all G ? frak FGin frak {F}. Then frak Xfrak {X} is called frak Ffrak {F}-normal. If frak F=frak S_pfrak {F}=frak {S}_{pi }, it is known that (1) frak Xfrak {X} is frak Ffrak {F}-normal precisely when frak X^*=frak F^*frak {X}^{ast }=frak {F}^{ast }, and consequently (2) frak F í frak Xfrak Nfrak {F}subseteq frak {X}frak {N} implies frak X^*=frak F^*frak {X}^{ast }=frak {F}^{ast }, and (3) there is a unique smallest frak Ffrak {F}-normal Fitting class. These assertions are not true in general. We show that there are Fitting classes frak Fnot = frak S_pfrak {F}not =frak {S}_{pi } filling property (1), whence the classes frak S_pfrak {S}_{pi } are not characterized by satisfying (1). Furthermore we prove that (2) holds true for all Fitting classes frak Ffrak {F} satisfying a certain extension property with respect to wreath products although there could be an frak Ffrak {F}-normal Fitting class outside the Lockett section of frak Ffrak {F}. Lastly, we show that for the important cases frak F=frak Nⁿ, ngeqq 2frak {F}=frak {N}^{n}, ngeqq 2, and frak F=frak S_p1?frak S_pr, p_i frak {F}=frak {S}_{p_{1}}cdots frak {S}_{p_{r}}, p_{i} primes, there is a unique smallest frak Ffrak {F}-normal Fitting class, which we describe explicitly.