On a class of groups whose normal subgroups are characteristic (Q1576998)

Let \(H\) be a subgroup of the group \(G\). The authors consider the action of \(N_G(H)/C_G(H)\) on the subgroups of \(H\). The main result is concerned with soluble \(N/C\)-up groups, they satisfy the following condition: If \(K\subset H\) is normal in \(N_G(H)\), then it is characteristic in \(H\). The authors state necessary conditions for a soluble group to be an \(N/C\)-up group (Theorem 3.5) and show that they are not sufficient.