Let $Y$ be a subset of a topological space $X$. We say that a clopen subset $L$ of $Y$ lifts to $X$ whenever there exists a clopen subset $H$ of $X$ such that $H\cap Y=L$.

Let $X$ be a compact and $T_0$-space and $Y$ be a closed subset of $X$. I am looking for conditions under which the clopen subsets of $Y$ lift to $X$. For example, if $Y$ is clopen then the clopen subsets of $Y$ lift to $X$.