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 prove that every finite soluble group satisfies $\ell_{\mathrm W}(G)\le h(G)\le4\ell_{\mathrm W}(G)$, where $h(G)$ is the Fitting height, and that $h(G)\le3\ell_{\mathrm W}(G)$ if $|G|$ 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 $n\le h\le4n$ occurs as $(\ell_{\mathrm W}(G),h(G))$. The constant $4$ comes from primitive groups. A soluble primitive group $V\rtimes M$ satisfies $\mathrm W(V\rtimes M)=V\rtimes M$ 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 $\mathrm{Out}(Q_8)\cong S_3$. We also determine when the base group of a soluble regular wreath product is the first term of the series.