Groups with a character of large degree relative to a normal subgroup

Let N be a normal subgroup of a finite group G and θ be an irreducible character of N which is fixed by the conjugation action of G. Let χ be an irreducible character of G that restricts to a multiple of θ on N. Then d=χ(1)/θ(1) is an integer which divides |G:N| and has |G:N|≥d2. We can thus write |G/N|=d(d+e) for a non-negative integer e and ask what can be said about d and G/N for a given e.This is a generalization of a problem considered by Snyder [12] where he takes d to an irreducible character degree of G and writes |G|=d(d+e). Berkovich has shown in [1] that if e=1, then G is a sharply 2-transitive group. For e≠1, Snyder shows in [12] that d is bounded by a function of e. This bound is later improved by Isaacs in [7] and then by Durfee and Jensen in [2] and Lewis in [9]. In this more general version of the problem, we will work under the assumption that G/N is solvable. We will show that for e>0, if d>(e−1)2 then e divides d and d/e+1 is a prime power. If in addition, either d>e5−e, (d/e,e)=1, or (d/e+1,e)=1 then there exist groups X, Y with N⊆X◁Y⊆G such that Y/X is a sharply 2-transitive group of order (d/e)(d/e+1).