Consider the cycle index of $S_N$ \begin{equation} \frac{1}{N!} \sum_{g} a_1^{j_1(g)}...a_N^{j_N(g)} \end{equation} Consider now the following generalization, where we insert two characters in the cycle index \begin{equation} \frac{1}{N!} \sum_{g} \chi^h(g) \chi^{h'}(g) a_1^{j_1(g)}...a_N^{j_N(g)} \end{equation} where $\chi^h(g)$ is a character of $S_N$ in an irreducible representation labelled by a conjugacy class $h$.
Has such objects been studied in group theory (any references would be helpful)? If so, what is being counted?