Let $\chi_1,...,\chi_N$ be the irreducible characters of a finite group $G$. Also, suppose $\chi_1,...\chi_m$, with $m < N$, are all the real irreducible characters. And $\chi_{m+1}, \overline{\chi_{m+1}},...,\chi_{n},\overline{\chi_{n}}$ are the remaining irreducible characters (each non-real). So that means $m + 2(n-m) = N$. Is it true that $\{\chi_1,...,\chi_m\} \cup\{\chi_{m+1} + \overline{\chi_{m+1}},...,\chi_{n}+\overline{\chi_{n}}\}$ is a basis for $$\{f:G \to \mathbb{C}:f \text{ is constant on }C_g \cup C_{g^{-1}} \text{ for }g \in G\}$$ where $C_g$ is the conjugacy class of $g$?
If so, can this be proven directly (showing the set is spanning)? I would like to avoid using results relating real conjugacy classes and real irreducible characters.
I would appreciate a hint rather than a full solution. Thank you.