Size of conjugacy class need not divide index of abelian normal subgroup

Statement
It is possible to have a finite group $$G$$, a conjugacy class $$c$$ of $$G$$, and an abelian normal subgroup $$H$$ of $$G$$ such that the size of $$c$$ does not divide the index $$[G:H]$$.

This puts a non-constraint on the sizes of conjugacy classes.

Example of symmetric group of degree three
In symmetric group:S3, a group of order six, there is a subgroup A3 in S3 that is abelian, normal, and has index two.