At page (28) of chapter I of the book Finite Group Theory by I.Martin.Issacs, one finds:
Let $G$ be a finite group, with Frattini subgroup $\Phi$. If $\Phi \subseteq N \vartriangleleft G$, and if $N/ \Phi$ is nilpotent, then $N$ is nilpotent.
I know that this is in fact a generalisation of two preceding exercises, but I could not prove it: I have tried to construct a central series of $N$ from that of $N/ \Phi$, but failed to do so. I also tried in the direction of showing that maximal subgroups of $N$ are normal in $N$, but thus far found nothing interesting. As this is a generalisation of one exercise which deploits of the Frattini arguments, I wanted to avail myself of that argument as well, while finding nothing critical either. Therefore I post here for some help.