Number of proper vertex colorings where a subgraph of a graph has a fixed coloring

Let $$G$$ be a finite simple graph, and let $$H$$ be a subgraph of $$G$$. Let $$q\geq 1$$ be an integer and suppose that $$P_G(q)$$ is the number of proper vertex $$q$$-colorings of $$G$$ (i.e.,proper vertex colorings of $$G$$ using at most $$q$$ colors).

I have been thinking about the number of such $$q$$-colorings, where $$H$$ has a fixed $$q$$-coloring. I wonder, if known, whether this number establishes a relationship between $$P_H(q)$$ and $$P_G(q)$$.

I have the feeling that it is not known in general and it is difficult to determine because when fixing a coloring of $$H$$, in order to form a coloring of $$G$$, one must carefully consider how to color the neighbors of the vertices of $$H$$ that are not in $$H$$. Nonetheless, I thought it was worth asking and if someone could refer me to a reference. Thank you.

If you fix a $$q$$ coloring on some node of the graph, then you can reduce to a classical coloring problem with the following transformation:
Let $$A$$ be the set of vertices already colored, and $$B$$ the vertices that are not. We build a graph $$G'$$ where the vertices are $$B\cup\{v_1, v_2, ..., v_q\}$$. All vertices of $$\{v_1, v_2, ..., v_q\}$$ are linked together as a complete graph. Two vertices of $$B$$ are linked together if and only if they are linked in $$G$$. If $$u\in B$$ and $$v_i \in \{v_1, v_2, ..., v_q\}$$, then $$uv_i$$ is an edge if and only if $$u$$ is adjacent in $$G$$ to a vertex with color $$i$$.
If $$H$$ is big, it can significantly reduce the size of the graph to color, but in general, it will no make the question easier. For example, if $$H$$ is reduced to a single vertex, we easily see we do not learn anything about $$G$$.
If you really want a recurrence formula, the deletion-contraction formula gives a recurrence linking the number of colorings of $$G$$ to the number of colorings of a subgraph of $$G$$ and a minor of $$G$$