Distinct characters with the same character values, outer automorphisms and Galois conjugation

Given an (irreducible complex) character of a finite group the following three construction all yield another irreducible character of the same degree:

1. multiplying by a degree 1 character
2. applying an outer automorphism
3. taking a Galois conjugate

Note that 2) and 3) never change the set of character values, they just permute the list.

The degree five characters of $ A_6 $ are not Galois conjugate but are related by an outer automorphism. The two degree $ 16 $ characters of $ M_{11} $ are not related by an outer automorphism but are Galois conjugate.

What is an example of a finite group $ G $ and two distinct irreducible characters of $ G $ which have the same character values but are not related by any combination of the three constructions given above?

This is in some sense a follow up question to

Same character values iff related by outer automorphism, for perfect groups