On a difference between $i_!$ and $i_*$ over $\mathbb{P}^1$

Question

Let $X$ be a smooth projective surface in $\mathbb{P}^3$ containing a line $l$. Denote by $C$ the curve corresponding to the divisor $2l$. Let $p \in C$ be a closed point. Denote by $U:=C \backslash p$ and $i:U \hookrightarrow C$, the natural immersion. There is a natural injective morphism $\pi$ from $\mathcal{H}om_C(\mathcal{I}_C/\mathcal{I}_C^2,i_!\mathcal{O}_C|_U)$ to $i_*(\mathcal{H}om_U(i^*(\mathcal{I}_C/\mathcal{I}_C^2),\mathcal{O}_U))$ given by restriction of morphisms from $C$ to $U$. Then what is the cokernel of $\pi$? Is the morphism from $H^0(i_*(\mathcal{H}om_U(i^*(\mathcal{I}_C/\mathcal{I}_C^2),\mathcal{O}_U)))$ to $H^0(\mbox{coker } \pi)$ surjective? Note that here $i_!$ is the extension by zero as mentioned in Hartshorne Ex II.$1.19$. By $\mathcal{I}_C$, I mean the ideal sheaf of $C$ in $\mathbb{P}^3$.