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Let $ G $ be a finite perfect group. Let $ \chi_1, \chi_2 $ be two different irreducible characters of $ G $. Suppose that the set of values taken by $ \chi_1 $ is the same as the set of values taken by $ \chi_2 $. Then must $ \chi_1 $ and $ \chi_2 $ be related by an outer automorphism of $ G $?
In the examples I know this is always true. For example the two degree $ 2 $ characters of $ SL(2,5) $ take the same values and are indeed related by an outer automorphism of $ SL(2,5) $.
There are plenty of counterexamples. The ATLAS of finite groups is a good place to look. I think the smallest is for the group ${\rm PSL}(2,11)$, for which there are two characters of degree $12$ with the same values, and they are not interchanged by the outer automorphism.
Another example is $M_{11}$ which has two degree $10$ characters and also two degree $16$ characters with the same values, and $M_{11}$ has trivial outer automorphism group.
A typical reason for this is that characters that arise as Galois conjugates often have the same values, and they may or may not be related by a group automorphism.
