Automorphism $\pi(\omega) \in \rm{Gal}(k(p \mathcal O_K) / \mathcal F_p )$ corresponding to the unique nontrivial $\omega \in \rm{Gal}(K / \mathbb Q)$

Question

Let $K$ be a quadratic number field and let $p$ be an inert prime, hence $k(p \mathcal O_K) / \mathcal F_p$ has degree 2.

In addition, we have that $\pi : \operatorname{Gal}(K / \mathbb Q) = \operatorname{Gal}(k ( p \mathcal O_K)/\mathcal F_p)$.

How can we compute the automorphism $\pi(\omega) \in \operatorname{Gal}(k(p \mathcal O_K)/\mathcal F_p)$ corresponding to the unique nontrivial $\omega \in \operatorname{Gal}(K/\mathbb Q)$ ?