Let $\Gamma$ be a congruence subgroup of $\text{SL}_2(\mathbb{Z})$ of level $N$ and let $\pi:\text{SL}_2(\mathbb{Z})\to\text{SL}_2(\mathbb{Z}/N\mathbb{Z})$ be the reduction map, which is surjective. Let $$ C_N=\bigg\{\begin{pmatrix}x\\y \end{pmatrix}\in(\mathbb{Z}/N\mathbb{Z})^2\colon\text{$x$ and $y$ generate $\mathbb{Z}/N\mathbb{Z}$}\bigg\}.$$ Set $H=\pi(\Gamma)$. Then there is a natural $H$-action on $C_N$, and a natural bijection $\Phi:\text{Cusp}(\Gamma)\to H\backslash C_N$ between cusps of $\Gamma$ and $H$ orbits in $C_N$.
Normally for a cusp $\mathfrak{c}\in\text{Cusp}(\Gamma)$, set $H_{\mathfrak{c}}=\gamma_{t}^{-1}\Gamma\gamma_{t}\cap\text{SL}_2(\mathbb{Z})_\infty$, where $\gamma_t\in\text{SL}_2(\mathbb{Z})$ such that $\gamma_t\infty=t\in\mathfrak{c}$ and $\text{SL}_2(\mathbb{Z})_\infty$ is the stabilizer of $\infty$. Then the width of the cusp is defined as the index of $\{\pm\}H_{\mathfrak{c}}$ in $\text{SL}_2(\mathbb{Z})_\infty$.
Before going to lockdown, my professor told me that it is sometimes easier to compute these widths "in characterstic $N$", by which he meant using $C_N$ and $\text{SL}_2(\mathbb{Z}/N\mathbb{Z})$ and this is where I came up with. Note that by the Orbit stabilizer $\text{SL}_2(\mathbb{Z}/N\mathbb{Z})/\text{SL}_2(\mathbb{Z}/N\mathbb{Z})_\infty=\text{SL}_2(\mathbb{Z}/N\mathbb{Z})\cdot\infty\cong C_N$, so $\text{SL}_2(\mathbb{Z}/N\mathbb{Z})$ acts transitively on $C_N$.
Let $\mathfrak{c}$ be a cusp of $\Gamma$ and let $(x,y)$ be an element from the $H$-orbit corresponding to $\mathfrak{c}$. I choose $\gamma\in\text{SL}_2(\mathbb{Z})$ such that $\gamma(1,0)=(x,y)$. Then I define $\tilde{H}_{\mathfrak{c}}:=\gamma^{-1}H\gamma\cap\text{SL}_2(\mathbb{Z}/N\mathbb{Z})_\infty$. I have now two questions.
Is is indeed true that $\tilde{H}_{\mathfrak{c}}=\pi(H_{\mathfrak{c}})$? and how does $[\{\pm1\}\tilde{H}_{\mathfrak{c}}:\text{SL}_2(\mathbb{Z}/N\mathbb{Z})]$ relate to $[\{\pm 1\}H_{\mathfrak{c}}\colon\text{SL}_2(\mathbb{Z})_\infty]$?