On Intersections of System Normalizers

We give a negative answer to Problem 17.39 of the Kourovka Notebook. For every odd prime power $q$ and every $m\ge 1$, we construct a finite metabelian group $G_{m,q}$ with $Z_{\mathrm{\infty }}(G_{m,q})=1$ and $\mathrm{\mid }\mathrm{\Phi }(G_{m,q})\mathrm{\mid }=q$ for which the least number of system normalizers with trivial intersection is $m+1$. Thus the number is unbounded even when the order of the Frattini subgroup is fixed.