Coding nested mixing one-sided subshifts of finite type as Markov shifts having exactly the same alphabet

Let $X_{0}$, $X$ be mixing one-sided subshifts of finite type such that $X_{0}\subseteq X$. We show a necessary and sufficient condition for the existence of mixing Markov shifts $Y_{0}$, $Y$, $Y_{0}\subseteq Y$, and a conjugacy $\pi : Y\to X$ with $\pi (Y_{0})=X_{0}$, such that the sets of letters appearing in both systems are the same, more precisely, $L_{1}(Y_{0})=L_{1}(Y)$.