= 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> is
$$
\langle\chi_X,\chi_Y\rangle=\frac1{|G|}\sum_{g\in G}|X^g||Y^g|=\frac1{|G|}\sum_{g\in G}|(X\times Y)^g|.
$$
To 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>. Therefore
$$
\boxed{\langle\chi_X,\chi_Y\rangle=\#\bigl(G\backslash(X\times Y)\bigr).}
$$
The conjugation in the usual <character inner product> has no effect here because the permutation <characters> are real-valued.
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