In an unweighted bipartite graph $G = (X + Y,E)$, we would like to find a maximum matching, but among all those maximum matchings, we would like to find one that saturates a given subset $X_0\subseteq X$.
A necessary condition for the existence of such a matching is that $X_0$ satisfies Hall's marriage condition, i.e., for every $X'\subseteq X_0$, the number of neighbors of $X'$ is at least $|X'|$. If this condition is satisfied, we can find a matching $M$ that saturates all vertices of $X_0$, but $M$ is not necessarily a maximum matching in $G$.
Is it always possible to extend $M$ into a maximum matching? Alternatively, is there a different way to find a maximum matching that saturates $X_0$ when the necessary condition is satisfied?