Let $Y$ be a subset of a topological space $X$, we say that clopen subset of $Y$ lift to $X$ whenever $L$ is a clopen subset of $Y$ then 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 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 $.