In this paper, we study the structure of shift preserving operators acting on shift-invariant spaces in $L^{2}(G)$, where $G$ is a locally compact Abelian group. We generalize some results related to shift-preserving operator and its associated range operator from $L^{2}(\mathbb{R}^{d})$ to $L^{2}(G)$. We investigate the matrix structure of range operator $R(\xi) $ on range function $J$ associated to shift-invariant space $V$, in the case of a locally compact Abelian group $G$. We also focus on some properties like as normal and unitary operator for range operator on $L^{2}(G)$. We show that shift preserving operator $U$ is invertible if and only if fiber of corresponding range operator $R$ is invertible and investigate the measurability of inverse $R^{-1}(\xi)$ of range operator on $L^{2}(G)$.