Character values up to permutation of characters need not determine automorphism class

Statement
It is possible to have a finite group $$G$$ and a splitting field $$K$$ for $$G$$ in characteristic zero such that the following holds: There are elements $$g_1,g_2 \in G$$ in different conjugacy classes and a permutation $$\sigma$$ on the set of characters of irreducible representations (up to equivalence) of $$G$$ over $$K$$ such that the following holds:


 * For every irreducible character $$\chi$$, we have $$\chi(g_1) = \sigma(\chi)(g_2)$$.
 * $$g_1$$ and $$g_2$$ are not in the same automorphism class, i.e., they are not in the same orbit under the action of the automorphism group $$\operatorname{Aut}(G)$$ on $$G$$.

Similar facts

 * Character value need not determine similarity class of image under irreducible representation

Opposite facts

 * Character orthogonality theorem
 * Column orthogonality theorem
 * Character determines representation in characteristic zero
 * Splitting implies characters separate conjugacy classes

Example of the dihedral group
Note that the element $$a$$ of order four and the element $$x$$ of order two have the same character values up to permutation of characters, but they cannot be in the same orbit under the action of the automorphism group because the elements have different orders.