The concepts of d- and nd-Frattini chief factors of a finite group are introduced. Their ingrainment into that of the extended Frattini dual subgroup becomes the natural dual to Frattini and supplemented chief factors. Not only does a dual of the strengthened form of the Jordan–Hölder theorem arise, but also the $p$-nilpotent radical becomes the intersection of the centralizers of the nd-Frattini chief factors. As a result, a class $\mathfrak{F}$ of groups is a full integrated local formation $\mathrm{LF}(f)$ if and only if each nd-Frattini chief factor in $G\in\mathfrak{F}$ is $f$-central.