A question about normal sets in a finite subgroup

Question

Let $G$ a finite non-abelian group and suppose that $N$ is a normal subset (not a subgroup) of $G$, that is, $aN=Na$ for every $a\in G$. Under what conditions $|N|$ divides $|G|$ ?

Answer 1

$N$ is necessarily the union of entire conjugacy classes. The size of every conjugacy class divides $|G|$ (by the orbit stabiliser theorem) so if $N$ consists of only one conjugacy class, its size will divide the group order. If it contains more than one conjugacy class, not much can be said about divisibility in general. For example, in $S_n$ you could make $N$ the union of any set of classes of all permutations with certain cycle-types. So in $S_4$ the size of $N$ could be any of 1,3,8,6,6 or any of their sums.