Let G be a simple undirected graph and let A be an additive Abelian group with identity 0. In this paper, we introduce the concept of group magic spectrum of a graph G with respect to a given Abelian group A and is defined as spec(G, A):= {λ : λ is a magic constant of some A-vertex magic labeling f }. In their recent work, K. M. Sabeel et al. in Australas. J. Combin. 85(1) (2023), 49-60 proved a forbidden subgraph characterization for the group vertex magic graph. In this work, we present a new method which uses minimum number of vertices required for this graph. We obtain a necessary and sufficient condition for the spectrum of a graph G to be a subgroup when A = V4 or Zp, where p is a prime number. Also we introduce the notion of reduced spectrum redspec(G, A) and study the relation between spec(G, A) and redspec(G, A).