Let $V_k(\mathbb{R}^n)$ be Stiefel manifolds.

In the paper *The cohomology rings of real Stiefel manifolds with integer coefficients*, Martin Čadek, Mamoru Mimura, and Jiří Vanžura, J. Math. Kyoto Univ.
Volume 43, Number 2 (2003), 411-428.,
the cohomology rings
$$
H^*(V_k(\mathbb{R}^n);\mathbb{Z}_2),
$$
$$
H^*(V_k(\mathbb{R}^n);\mathbb{Z}),
$$
are obtained.

Let $\Sigma_k$ be permutation grup of order $k$. For any $\sigma\in \Sigma_k$, let $\sigma$ act on $V_k(\mathbb{R}^n)$ by $$\sigma(v_1,\cdots,v_k)=(v_{\sigma(1)},\cdots,v_{\sigma(k)}).$$

I want to know the induced homomorphism on cohomology ring $$ \sigma^*: H^*(V_k(\mathbb{R}^n);\mathbb{Z}_2)\to H^*(V_k(\mathbb{R}^n);\mathbb{Z}_2), $$ $$ \sigma^*: H^*(V_k(\mathbb{R}^n);\mathbb{Z} )\to H^*(V_k(\mathbb{R}^n);\mathbb{Z} ). $$ Is it possible or difficult? How to solve it?

Could you illustrate the example how to obtain the action of $\mathbb{Z}_2$ on $H^*(V_2(\mathbb{R}^n);\mathbb{Z})$?