This question is related to this other one

A Schur positivity conjecture related to row and column permutations

by Richard Stanley (thanks to Sam Hopkins for letting me know about it).

Consider a Young subgroup $S_{\lambda}$ of the symmetric group $S_n$, corresponding to some integer partition $\lambda$ of $n$. Let $\tau$ be some permutation and define the symmetric function

$$ F(\tau)=\sum_{\sigma\in S_{\lambda}}p_{c(\tau\sigma)} $$ where $p_{\mu}$ is the usual power sum symmetric function and $c(\rho)$ denotes the integer partition given by the cycle-type of the permutation $\rho$.

**Q:** What is known about the Schur function expansion of $F(\tau)$, given the double coset class of $\tau$ for the Young subgroup?