19.82 (2018)

Let $h^*(G)$ denote the generalized Fitting height of a finite group $G$ defined as the minimum number $k$ such that $F^*_k(G) = G$, where $F^*_1(G) = F^*(G)$ is the generalized Fitting subgroup of $G$, and by induction $F^*_{i+1}(G)$ is the inverse image of $F^*(G/F^*_i(G))$. If $G$ is soluble, then $h^*(G) = h(G)$ is the Fitting height of $G$. Does every finite group $G$ contain a soluble subgroup $K$ such that $h^*(G) = h(K)$?