Given a graph $G(V,E)$ and a partition $\{V_1,\dots V_n\}$ of the nodes set $V$, the adjacency and Laplacian spectra of the quotient graph $Q(G)$ interlaces the adjacent and the Laplacian spectra of the parent graph $G$. Besides, if the partition in equitable the interlacing is tight

This is a well known result.

Now, suppose I have a directed graph $D(V,E)$ and a partition $\{V_1,\dots V_n\}$ of the nodes set. Suppose also that the quotient graph $Q(D)$ is undirected. Under what conditions the interlacing and inclusion results for the adjacency and/or Laplacian spectra are still valid?