Multiplying a conjugacy class by its inverse in a finite group

Suppose that G is a finite group and K a non-trivial conjugacy class of G such that KK⁻¹=1∪ D∪ D⁻¹ with D a conjugacy class of G. We prove that G is not a non-abelian simple group. We also give arithmetical conditions on the class sizes determining the structure of K and D. Furthermore, if $D=K$ is a non-real class, then K is p-elementary abelian for some odd prime p.