Finite group in which all cumulative conjugacy class size statistics values divide the order of the group

Definition
A finite group in which all cumulative conjugacy class size statistics values divide the order of the group is a finite group $$G$$ with the following property: for every natural number $$n$$, the (number of elements in $$G$$ for which the size of the conjugacy class of that element divides $$n$$) divides the order of $$G$$.

To evaluate this property for a group, it suffices to know the conjugacy class size statistics.

Examples
In addition to all finite abelian groups, all groups of order $$p^3$$, $$p^4$$, and $$p^5$$ have this property for any prime number $$p$$.

However, there exist counterexamples of order $$p^6$$ for every prime number $$p$$, so the property is not held by every finite nilpotent group or even by every group of prime power order.

The smallest counterexample is symmetric group:S3, which is a group of order 6 where the number of elements in conjugacy classes of size dividing 3 is 4.