A Character Condition for Quadruple Transitivity

Let G be a permutation group of degree n viewed as a subgroup of the symmetric group S≅Sn. We show that if the irreducible character of S corresponding to the partition of n into subsets of sizes n−2 and 2, that is, to say the character often denoted by χ(n−2,2), remains irreducible when restricted to G, then n = 4, 5 or 9 and G≅S3, A5, or PΣL2(8), respectively, or G is 4-transitive.