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 ${\mathrm{\ell }}_{\mathrm{W}}(G)$ be the length of the associated upper series. Thus $\mathrm{W}(G)=G$ exactly when all these residuals are normal in $G$. We prove that every finite soluble group satisfies ${\mathrm{\ell }}_{\mathrm{W}}(G)\le h(G)\le 4{\mathrm{\ell }}_{\mathrm{W}}(G)$, where $h(G)$ is the Fitting height, and that $h(G)\le 3{\mathrm{\ell }}_{\mathrm{W}}(G)$ if $\mathrm{\mid }G\mathrm{\mid }$ is odd or the Sylow $2$-subgroups of $G$ are abelian. Both constants are attained at every length, the constant $4$ by $\{2,3\}$-groups. Moreover, every pair $(n,h)$ with $n\le h\le 4n$ occurs as $({\mathrm{\ell }}_{\mathrm{W}}(G),h(G))$. The constant $4$ comes from primitive groups. A soluble primitive group $V⋊M$ satisfies $\mathrm{W}(V⋊M)=V⋊M$ if and only if every abelian subgroup of $\mathrm{F}(M)$ is cyclic. It then has Fitting height four if and only if $\mathrm{F}(M)=C\times Q_8$, with $C$ cyclic, and $M$ induces the full group $\mathrm{O}\mathrm{u}\mathrm{t}(Q_8)\cong S_3$. We also determine when the base group of a soluble regular wreath product is the first term of the series.