It is widely known that $$ \frac{1}{n!}\sum_{\pi\in S_n}\chi_\lambda(\pi)\chi_\mu(\pi)=\delta_{\lambda,\mu},$$ where $S_n$ is the permutation group and $\chi$ are its irreducible characters.

In exercise 7.63 of his classic book *Enumerative Combinatorics*, Richard Stanley computes explicitly the value of
$$\sum_{\pi\in D_n}\chi_\lambda(\pi),$$
where $\lambda$ is a hook and $D_n$ is the set of derangements (permutations without fixed points).

I would like to know the value of $$ \sum_{\pi\in D_n}\chi_\lambda(\pi)\chi_\mu(\pi),$$ at least when $\lambda$ and/or $\mu$ is a hook. Is anything known about this sum? (It is a generalization of the previous one, to which it reduces when $\mu=(n)$).