Orbit sizes for irreducible representations may differ from orbit sizes for conjugacy classes under action of automorphism group

Statement
It is possible to have a finite group $$G$$ with the property that under the action of the automorphism group $$\operatorname{Aut}(G)$$, the orbit sizes for the set $$R(G)$$ of irreducible linear representations of $$G$$ are not the same as the orbit sizes for the set $$C(G)$$ of conjugacy classes of $$G$$.

A finite group where the orbit sizes are in fact the same is termed a finite group having the same orbit sizes of conjugacy classes and irreducible representations under automorphism group.

Opposite facts
The most direct opposite fact is: Cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group‎

This gives a sufficient condition for being a finite group having the same orbit sizes of conjugacy classes and irreducible representations under automorphism group. The following hold for all finite groups:


 * Number of irreducible representations equals number of conjugacy classes
 * Number of orbits of irreducible representations equals number of orbits under automorphism group
 * Number of irreducible representations over reals equals number of equivalence classes under real conjugacy
 * Number of irreducible representations over rationals equals number of equivalence classes under rational conjugacy
 * Number of irreducible representations over complex numbers with real character values equals number of conjugacy classes of real elements

Similar facts

 * Number of irreducible representations over rationals need not equal number of conjugacy classes of rational elements

Proof
There are examples of groups of order 27.