Squares of real conjugacy classes in finite groups

We prove that if a finite group G contains a conjugacy class K whose square is of the form 1 D, where D is a conjugacy class of G, then K is a solvable proper normal subgroup of G and we completely determine its structure. We also obtain the structure of those groups in which the assumption above is true for all non-central conjugacy classes and when every conjugacy class satisfies that its square is the union of all central conjugacy classes except at most one.