Let $V=\mathbb{A}_{\mathbb{C}}^n/G$ be an quotient singularity where $G=\mathbb{Z}/r\mathbb{Z}$ and $G$ acts on $\mathbb{A}^n$ by $(x_1,\ldots,x_n) \mapsto (\zeta^{a_1}x_1,\ldots,\zeta^{a_n}x_n)$ and $(a_1,\ldots,a_n)=1$.

Let $0 \neq f \in \mathbb{C}[z_1,\ldots,z_n]$ be a $G$-eigenfunction and $X=(f=0)/G \subset V$ be the Weil divisor of $V$. Let $I \subset \mathcal{O}_V$ be the ideal sheaf of $X$ in $V$. There is an exact sequence

$ I/I^2 \rightarrow \Omega^1_V|_X \rightarrow \Omega^1_X \rightarrow 0. $

I want to know whether the map $I/I^2 \rightarrow \Omega^1_V|_X$ is injective or not in general. I think this is well known if $r=1$ i.e. l.c.i. case.