Let $X$ be a projective connected scheme embedded into $\mathbb{P}^n$ for $n \ge 3$. Denote by $i$ the natural inclusion morphism from $X$ into $\mathbb{P}^n$. Denote by $X_H$ the intersection of $X$ with $H$, a general hyperplane in $\mathbb{P}^n$. Denote by $i_H:X_H \hookrightarrow \mathbb{P}^n$ the natural inclusion. Is it true that for a general hyperplane $H$ in $\mathbb{P}^n$, any $\mathcal{O}_{\mathbb{P}^n}$-morphism, $\mbox{Hom}_{\mathbb{P}^n}(\mathcal{I}_X,i_{H_*}\mathcal{O}_{X_H})$ be extended to an element in $\mbox{Hom}_{\mathbb{P}^n}(\mathcal{I}_X,i_*\mathcal{O}_{X})$? In other words, is the natural morphism from $\mbox{Hom}_{\mathbb{P}^n}(\mathcal{I}_X,i_*\mathcal{O}_{X})$ to $\mbox{Hom}_{\mathbb{P}^n}(\mathcal{I}_X,i_{H_*}\mathcal{O}_{X_H})$ surjective for a general choice of $H$?
EDIT Assume $X$ is local complete intersection in $\mathbb{P}^n$