Wielandt subgroups of nilpotent residuals and Fitting-height bounds

For a finite group $G$, let $\mathrm W(G)$ be the intersection of the normalizers of the nilpotent residuals of the subnormal subgroups of $G$, and let $\ell_{\mathrm W}(G)$ be the length of the associated upper series. Thus $\mathrm W(G)=G$ exactly when all these residuals are normal in $G$. We p