Let \(G\) be a finite group and order the set of sizes of conjugacy classes of \(G\) decreasingly to obtain what is called the conjugate type vector of \(G\). The authors show by examples that if \(H\) is nilpotent and if \(G\) and \(H\) have the same conjugate type vector, then \(G\) is not necessarily nilpotent. This remark leads the authors to consider the A-groups (finite groups with all Sylow subgroups Abelian). They prove the following two results, of which the first seems to this reviewer most elegant: 1) Let \(G\) be an A-group having a conjugacy class of size \(2^n\), where \(2^n\) is the maximum power of \(2\) dividing the conjugacy class size of any element of \(G\). Then \(G\) is solvable. 2) If \(G\) is a metabelian A-group with a conjugate type vector like that of a nilpotent group, then \(G\) is Abelian.