  A group $G$ is said to be totally $2$-closed if in each of its faithful
permutation representations, say on a set $\Omega$, $G$ is the largest subgroup
of $\mathrm{Sym}(\Omega)$ which leaves invariant each of the $G$-orbits for the
induced action on $\Omega\times \Omega$. We prove that there are precisely $47$
finite totally $2$-closed groups with trivial Fitting subgroup. Each of these
groups is a direct product of pairwise non-isomorphic sporadic simple groups,
with the direct factors coming from the Janko groups $\mathrm{J}_1,
\mathrm{J}_3$ and $\mathrm{J}_4$, together with $\mathrm{Ly}, \mathrm{Th}$ and
the Monster $\mathbb{M}$. These are the first known examples of insoluble
totally $2$-closed groups. As a by-product of our methods, we develop several
tools for studying $2$-closures of transitive permutation groups -- a vital
tool in the study of representations of finite groups as automorphism groups of
digraphs. We also prove a dual to a 1939 theorem of Frucht from Algebraic Graph
Theory.

Group Theory;Combinatorics