Given an integer $N>0$, not necessarily prime, we have the paramodular group $K(N) \subset \text{Sp}_{4}(\mathbb{Q})$, which consists of matrices of the form
$$\begin{bmatrix} * & *N & * & *\\ * & * & * & \frac{*}{N} \\ * & *N & * & * \\ *N & *N & *N & * \end{bmatrix} \in \text{Sp}_{4}(\mathbb{Q}) \,\,\,\,\,\,\,\,\,\,\,\,\, * \in \mathbb{Z}$$
We define the group $K^{*}(N) = \langle K(N), V_{N} \rangle$ generated by the paramodular group, and the matrix
$$V_{N} = \frac{1}{\sqrt{N}}\begin{bmatrix} 0 & N & 0 & 0\\ -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -N & 0 \end{bmatrix}$$
You can equivalently generate $K^{*}(N)$ with the parabolic subgroup isomorphic to $\text{SL}_{2}(\mathbb{Z}) \rtimes \mathbb{Z}^{2}$ and $V_{N}$, i.e. $K^{*}(N) = \langle \text{SL}_{2}(\mathbb{Z}) \rtimes \mathbb{Z}^{2}, V_{N} \rangle$.
So I have a function which transforms correctly under the very similar group $\langle \Gamma_{0}(N) \rtimes \mathbb{Z}^{2}, V_{N} \rangle$. I'm wondering, is this a common group? And does it have a moduli interpretation in abelian surfaces? I thought that replacing $\text{SL}_{2}(\mathbb{Z})$ by $\Gamma_{0}(N)$ might correspond to abelian surfaces with partial level structure and polarization $(1,N)$?
I think what I'm after is close to the Iwahori subgroup (along with $V_{N}$). But not quite. The Iwahori subgroup is $B(N) = \Gamma_{0}^{(2)}(N) \cap K(N)$, where $\Gamma_{0}^{(2)}(N)$ is the obvious Siegel congruence subgroup. The group $\langle B(N), V_{N} \rangle$ is close, but we lose that one non-integral entry $\frac{*}{N}$ in the paramodular group. And that entry seems to be important for me.
