Let $G$ be a unimodular locally compact group, $N,A \le G$ be unimodular closed subgroups. Suppose that $A$ normalizes $N$. Let $N_0 \le N$ be a compact open subgroup. Suppose that a function $f : N \rightarrow \mathbb{C}$ is supported on $\bigcup_{a \in A} a N_0 a^{-1} $. Is there a formula to compute $$ \int_N f(x) dx $$in this case in terms of integration over $A$ and $N_0$? For example, if $G$ is a finite group, we have $$ \sum_{x \in N} f\left( x \right) = \sum_{a \in A} \sum_{x \in N_0} \frac{1}{\left| C_x \left( A \right) \right|} f\left( a x a^{-1} \right),$$ where $$C_x\left( A \right) = \left\{ a \in A \mid ax=xa \right\}$$is the centralizer of $x$ in $A$.

I am interested in particular in the case where $G$ is a matrix group (over a non-Archimedean field), $N$ is a unipotent radical of a parabolic subgroup of $G$, and $A$ is a subgroup of the Levi part of the same parabolic subgroup.