Let $p$ be an odd prime. Let $T^p=S^1\times\cdots \times S^1$ be the $p$-dimensional torus. Then $$H^*(T^p;\mathbb{Z}_p)=\otimes_pH^*(S^1;\mathbb{Z}_p)=\otimes_p\Lambda_{\mathbb{Z}_p}[a].$$ Here $|a|=1$.
Let $S_p$ be the permutation group of order $p$. Let $S_p$ act on $T^p$ by permuting the coordinates $$\sigma\in S_p: T^p\to T^p,$$ $$(x_1,\cdots,x_p)\mapsto (x_{\sigma(1)},\cdots,x_{\sigma(p)}).$$
How to get $$H^*(T^p/S_p;\mathbb{Z}_p)?$$