A problem of Grätzer and Wehrung on groups (Q1281269)

The authors answer a question in a paper by \textit{G.~Grätzer} and \textit{F.~Wehrung} [Acta Math. Hung. 85, No. 1-2, 175-185 (1999)]: Given a group \(G\), does there exist a proper supergroup~\(\overline G\) such that every normal subgroup \(N\) of \(G\) can be uniquely expressed in the form \(\overline N\cap G\) where \(\overline N\) is a normal subgroup of \(\overline G\). The elementary Abelian group of order \(p^2\) (for any prime \(p\)) is a counterexample to this conjecture since it is determined by its lattice of normal subgroups. But the authors show that for any group \(G\), the wreath product \(\overline G\) of a nonabelian simple group with \(G\) nearly has the desired property: the trivial subgroup is the only normal subgroup \(N\) of \(G\) for which there are (precisely) two normal subgroups \(\overline N\) of \(\overline G\) with \(N=\overline N\cap G\).