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) $.