On the intersection of maximal supersoluble subgroups of a finite group

The hyper-generalized-center $genz^*(G)$ of a finite group $G$ coincides with the largest term of the chain of subgroups $1=Q_0(G)\le Q_1(G)\le\ldots\le Q_t(G)\le\ldots$ where $Q_i(G)/Q_{i-1}(G)$ is the subgroup of $G/Q_{i-1}(G)$ generated by the set of all cyclic $S$-quasinormal subgroups of $G/Q_{i-1}(G)$. It is proved that for any finite group $A,$ there is a finite group $G$ such that $A\le G$ and $genz^*(G)\ne\text{Int}_\mathfrak{U}(G)$.