Let $D = \text{diag}(1,d)\in M_{2}(\mathbb{Z})$ be a $2\times 2$ matrix, where $d$ is an odd integer. We define the subgroup $\Gamma_D\subset M_{4}(\mathbb{Z})$ as: $$\Gamma_D := \left\lbrace R\in M_{4}(\mathbb{Z}) \: | \: R \left(\begin{matrix} 0 & D \\ -D & 0 \end{matrix}\right) R^t = \left(\begin{matrix} 0 & D \\ -D & 0 \end{matrix}\right) \right\rbrace,$$ and the subgroup $\Gamma_D(D)\subset\Gamma_D$ as: $$\Gamma_2(1,d):= \left\lbrace \left(\begin{matrix} a & b \\ c & d \end{matrix}\right)\in \Gamma_D \: | \: a-Id_2 \equiv b \equiv c \equiv d-Id_2 \equiv 0 \: mod(D) \right\rbrace,$$ where $A\equiv 0\: mod(D)$ means that $A = D\cdot B$ for some $B\in M_2(\mathbb{Z})$.

Does there exists an exact sequence

$$1 \to \Gamma_2(2,2d) \to \Gamma_2(1,d) \to Sp(4,\mathbb{Z}/2\mathbb{Z}) \to 1,$$

where the last map is the reduction $mod\ 2$ of the entries of the matrices? In order to prove this I wanted to check the index of the level subgroups inside $\Gamma_2$. I know that there exists a formula by Igusa for the level $n$ subgoups, and I wonder if also these indexes are known, or at least easily computable.

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