On the lattice of subgroups normalized by a symmetric one in the complete monomial group

We consider a lattice of subgroups normalized by a symmetric group $S_n$ in the complete monomial group $G=H\wr S_n$ where $H$ is an arbitrary (finite or infinite) group. It is shown that for $n\ge3$ the subgroup is strongly paranormal in this wreath product for any $H$. A similar result is obtained for an alternating group $A_n$, $n\ge4$. The property of strong paranormality for $D$ in $G$ means that for any element $x\in G$ the commutator identity $[[x,D],D]=[x,D]$ holds. That condition garantees a standard arrangement of subgroups of $G$ normalized by $D$.