A problem of Kegel (Q1316846)

Let \(\mathfrak X\) be a class of finite groups which is closed under subgroups, homomorphic images and extensions. A subgroup \(H\) of a finite group \(G\) is said to be \(\mathfrak X\)-normal in \(G\) if there exists a chain of subgroups \(H = H_ 0 \leq H_ 1 \leq \dots \leq H_ n = G\) such that either \(H_ i\) is normal in \(H_{i + 1}\) or \(H_{i + 1}/(H_ i)_{H_{i + 1}}\) belongs to \(\mathfrak X\) for every \(i < n\) (here \((H_ i)_{H_{i + 1}}\) denotes the core of \(H_ i\) in \(H_{i + 1}\)). Let \(G^{\mathfrak X}\) denote the \({\mathfrak X}\)-coradical of the finite group \(G\), i.e. the smallest normal subgroup of \(G\) such that \(G/G^{\mathfrak X}\) is an \(\mathfrak X\)-group. Answering a question of \textit{O. H. Kegel} [Arch. Math. 30, 225-228 (1978)], the author proves that if \(H\) is an \(\mathfrak X\)- normal subgroup of a finite group \(G\) and \(H = H^{\mathfrak X} = H'\), then \(HK = KH\) for every \(\mathfrak X\)-normal subgroup \(K\) of \(G\).