20.1 (2022)

For $k \geqslant 1$, a group $G$ is said to be totally $k$-closed if in each of its faithful permutation representations, say on a set $\Omega$, $G$ is the largest subgroup of $\text{Sym}(\Omega)$ which leaves invariant each of the $G$-orbits in the induced action on the set of ordered $k$-tuples $\Omega^k$. Are there any finite insoluble totally 2-closed groups with nontrivial Fitting subgroup?

The finite totally 2-closed groups which are either soluble or have trivial Fitting subgroup are known.