Cosets of normal subgroups and union of two conjugacy classes

Let G be a finite group, N a normal subgroup of G and $x\in G-N$. We discuss when the coset Nx is contained in the union of two conjugacy classes, K and D, of G. We show that N need not be solvable, and can even be non-abelian simple, but in these cases, K and D must have the same cardinality, and the non-solvable structure of N is restricted. The non-abelian principal factors of G contained in N are then isomorphic to $S\times \cdots \times S$, where S is a simple group of Lie type of odd characteristic.