Doubly transitive groups and lattices (Q1296164)

Let \(G\) be a finite group acting faithfully and doubly transitively on a finite set \(\Omega\). Then the corresponding permutation module \(\langle e_{x} \mid x \in {\mathbb C}\rangle\) decomposes as the sum of the trivial module and another irreducible constituent say \(V\). The author computes the automorphism groups of \(G\)-invariant integral lattices \(\Lambda\) contained in \(V\) for the case \(G\) has non-abelian socle. The result is that \[ {\mathbb Z}_{2} \times G \leq \Aut(\Lambda) \leq {\mathbb Z}_{2} \times \Aut(G), \] with an explicit list of a few exceptions. The proof uses the classification of finite simple groups, in particular, the classification of doubly transitive permutation groups.