  We prove that if a finite group $G$ contains a conjugacy class $K$ whose
square is of the form $1 \cup D$, where $D$ is a conjugacy class of $G$, then
$\langle K \rangle$ 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.
