 Solution

The character of a permutation representation at $g$ equals the number of points fixed by $g$, because a permuted basis vector contributes one to the trace exactly when it is fixed. Thus the character inner product isTo identify the last average, count pairs $(g,z)$ with $gz=z$. Each orbit contributes $|G|$, since its number of elements times their common stabilizer size is $|G|$ by the orbit-stabilizer theorem. ThereforeThe conjugation in the usual character inner product has no effect here because the permutation characters are real-valued.