Let \(G = (V , E)\) be a finite simple graph of order \(n\) and let \(\Gamma\) be an abelian group of order \(n\). A \(\Gamma\)-distance magic labeling of \(G\) is a bijection \(\varphi :V\rightarrow \Gamma\) for which there exits \(\gamma \in \Gamma\) such that \(\Sigma_{x \in N(V)} \varphi(x)=\gamma\) for any \(v \in V\), where \(N(v)\) is the neighbourhood of \(v\). Many people studied the existence of group magic labeling of various types of graphs, including direct products of cycles, Cartesian products, complete multipartite graphs, and Cayley graphs. In this paper the authors completely characterize all connected \(\Gamma\)- distance magic circulant graphs of valency four for any given finite abelian group \(\Gamma\) of order \(n\), based on some lemmas concerned about sequences over abelian groups. Their remarkable result is given below. Suppose \(n\geq6\) and \(\mathrm{Cay}(Z_n;\lbrace\pm a,\pm b\rbrace)\) is a connected tetravalent circulant graph. Let \(\Gamma\) be a finite abelian group of order \(n\) and let \(\Gamma_1\) be the subgroup of \(\Gamma\) generated by the elements of order 2. Let \(d = \gcd(n,a+b)\) and \(d^\prime = \gcd (n, a-b)\). Then \(\mathrm{Cay}(Z_n;\lbrace\pm a,\pm b\rbrace)\) is \(\Gamma\)-distance magic if and only if \(2|n\) and one of the following is satisfied: (1) \(2\nmid d\) and \((\frac{n}{2dd^\prime}) | \exp(\Gamma)\); (2) \(n = dd'\); (3) \(n = \frac{dd^\prime}{2}\) and one of \(d\), \(d^\prime\) is equal to \(2^k\), where \(k\geq2\) and \(2^{k-2} | |\Gamma_1|\).