2 corrected a typo

Consider $S_{n}$ the symmetric group and for each $\sigma\in S_{n}$ let $U_{\sigma}$ be its $n\times n$ permutation matrix. Let $A$ be an Hermitian $n\times n$ matrix. I'm interested in computing the average $$\mathbb{E}(A):=\sum_{\sigma \in S_{n}}{w(\sigma) U_{\sigma} A U_{\sigma}^{*}}$$

where the $w(\sigma)$ are some positive weight adding up to one.

For instance some natural weights are the ones coming from the Ewens's probability distribution of parameter $\theta>0$ on $S_{n}$ defined as

$$w(\sigma)=\frac{\theta^{K(\sigma)}}{\theta(\theta+1)\ldots(\theta+n-1)}$$