A question on irreducible characters.

Question

Let $G$ be a finite group. Let $Irr(G)$ be the set of irreducible characters of $G$ and let $Aut(G)$ be the automorphism group of $G$. It can be seen that $\chi^\beta \in Irr(G)$ if $\beta$ is an inner automorphism and $\chi^\beta(x)=\chi(\beta(x))$. In general is it true that $$\chi^\beta \in Irr(G) \text{ for all }\beta\in Aut(G)?$$

Answer 1

Irreducible characters cannot be written as sums of other characters.

Hint: $(\chi_1+\chi_2)^{\beta}=\chi_1^{\beta}+\chi_2^{\beta}$.

Answer 2

@runway44's answer is already complete (and should be the one accepted), but just to have an alternative answer:

For a character $\chi\in \mathrm{Irr}(G)$ it is known that $\chi$ is irreducible if and only if $$(\chi,\chi)=\frac{1}{|G|}\sum_{g\in G}\chi(g)\overline{\chi(g)}=1.$$ Use the definition of $\chi^\beta$ to compute $(\chi^\beta,\chi^\beta)$, under the assumption that $\chi$ is irreducible.