Let $(X_t)_{t \in \mathbb{N}}$ be a discrete-time Markov process on the finite state space $\mathcal{X}$, with transition matrix $T$. Suppose $f: \mathcal{X} \to \mathcal{Y}$ is a (deterministic) function such that $(f(X_t))_{t \in \mathbb{N}}$ is a Markov process on the state space $\mathcal{Y}$. What can we say about $f$ in relation to the transition matrix $T$ of the original chain?

Ideally, I would like an exact characterization of when the derived process is Markov. If $\mathcal{X} = \mathcal{Y} \times \mathcal{Z}$ and $f$ is projection onto the first component, then I think it suffices for the transition probability $T((y_1,z_1),(y_2,z_2))$ to be constant in $z_1$ **or** constant in $z_2$. There is also the case when $f$ is injective. Are there any other cases?