Subgroups which are the union of three conjugate classes (Q1270114)

Finite groups \(H\) are considered that can be embedded as normal subgroup in a group \(G\) such that \(H\) is the union of 3 conjugacy classes of \(G\). The results are well-known, see Theorem 2 of \textit{T. J. Laffey} and \textit{D. MacHale} [J. Aust. Math. Soc., Ser. A 40, 253-260 (1986; Zbl 0587.20016)]. The use of character theory here is unnecessary. For example, in Proposition 2, \(H\) is a \(p\)-group. As \(Z(H)\) and \(H'\) are characteristic subgroups of \(H\), it is immediate that \(H'=Z(H)\), so that \(H\) is even nilpotent of class two.