Let $p \neq 2$ and let $F$ be a $p$-adic field with ring of integers $\mathcal{O}$ and maximal ideal $\mathfrak{p}$.
By a quadratic space $V_{\mathcal{O}}$ of dimension $d$ over $\mathcal{O}$, I mean a free $\mathcal{O}$-module of rank $d$ endowed with a non-degenerate symmetric $\mathcal{O}$-bilinear form $B : V_{\mathcal{O}} \times V_{\mathcal{O}} \rightarrow {\mathcal{O}}$. The orthogonal group of $V$ is \begin{equation*} O(V_{\mathcal{O}}) = \lbrace f \in Hom_{\mathcal{O}}(V_{\mathcal{O}},V_{\mathcal{O}}) : B(f(v), f(w)) = B(v,w) \text{, } \forall v,w \in V_{\mathcal{O}} \rbrace. \end{equation*} Let $\vartheta \geq 1$ be an integer. The reduction map \begin{equation*} \mathcal{O} \rightarrow \mathcal{O}/\mathfrak{p}^{\vartheta} \end{equation*} modulo $\mathfrak{p}^{\vartheta}$ induces a map \begin{equation*} V_{\mathcal{O}} \rightarrow V_{\mathcal{O} / \mathfrak{p}^{\vartheta}} \end{equation*} where $V_{\mathcal{O} / \mathfrak{p}^{\vartheta}}$ is a free $\mathcal{O}/\mathfrak{p}^{\vartheta}$-module of rank $d$ endowed with the non-degenerate symmetric bilinear form $\overline{B} : V_{\mathcal{O} / \mathfrak{p}^{\vartheta}} \times V_{\mathcal{O} / \mathfrak{p}^{\vartheta}} \rightarrow {\mathcal{O} / \mathfrak{p}^{\vartheta}}$ defined by \begin{equation*} \overline{B}(\overline{v}, \overline{w}) = \overline{B(v,w)} \quad (v,w \in V_{\mathcal{O}}). \end{equation*} Define $O(V_{\mathcal{O}/ \mathfrak{p}^{\vartheta}})$ as the group of made up of $f \in Hom_{\mathcal{O}/ \mathfrak{p}^{\vartheta}}(V_{\mathcal{O} / \mathfrak{p}^{\vartheta}},V_{\mathcal{O} / \mathfrak{p}^{\vartheta}})$ such that \begin{equation*} \overline{B}(f(v), f(w)) = \overline{B}(v,w) \quad (v,w \in V_{\mathcal{O} / \mathfrak{p}^{\vartheta}}). \end{equation*}
Question: what do we know about the cardinal of $O(V_{\mathcal{O}/ \mathfrak{p}^{\vartheta}})$ or, equivalently, about the cardinal of $O(V_{\mathcal{O}}) / K_{\vartheta}$, where $K_{\vartheta}$ denote the kernel of the reduction map \begin{equation*} O(V_{\mathcal{O}}) \rightarrow O(V_{\mathcal{O}/ \mathfrak{p}^{\vartheta}}). \end{equation*} Remark: when $\vartheta = 1$ this is well-known and the cardinal of $O(V_{\mathcal{O}/ \mathfrak{p}})$ depends on the Witt index and the dimension ($0$, $1$ or $2$) of the "anisotropic" part of $V_{\mathcal{O}/ \mathfrak{p}}$ (see e.g. [Taylor - The Geometry of classical groups, Chapter 11])
