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

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.