$beta$- KMS States

Question

Let $(\mathcal{U},\tau)$ and $(\tilde{\mathcal{U}},\tilde{\tau})$ be two dynamical systems and let $\pi$ be a morphism between them. So I hope it should satisfy $\tilde{\tau}_t\circ \pi=\pi \circ \tau_t, \forall t \in \mathbb{R}$. Let $\tilde{\omega}$ be a $\beta$-KMS state for $(\tilde{\mathcal{U}},\tilde{\tau})$. Then if I define $\omega: \mathcal{U} \rightarrow \mathbb{C}$ by $\omega(a)=\tilde{\omega}(\pi(a))$, then will $\omega$ be a $\beta$-KMS state for $(\mathcal{U},\tau)$?