Let $U, V$ and $H$ be finite index subgroups of a group $G$ with $U$ and $V$ normal in $G$. Is there a way to relate representatives of $U/V$ with representatives of $H\cap U/H\cap V$?
Specifically, we are trying to relate the following formal sums:
$$\sum_{x_i \in U/V} x_i$$ $$\sum_{y_i \in H\cap U/H \cap V} y_i$$ in a way that rewrites the $y_i$ in terms of the $x_i$ or vice-versa.