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For cyclic LDPC codes, we propose to use their automorphism groups to improve the iterative decoding performance. The basic idea is to construct nonequivalent parity-check matrices via column permutations. Three types of iterative decoders are devised to take advantage of the code's automorphism group. In this paper we focus on cyclic LDPC codes defined by a circulant parity-check matrix and consider two known subgroups of the automorphism group of a cyclic code. For the large class of idempotent-based cyclic LDPC codes in the literature, we show that the two subgroups only provide equivalent parity-check matrices and thus cannot be harnessed for iterative decoding. Towards exploiting the automorphism group of a code, we propose a new class of cyclic LDPC codes based on pseudo-cyclic MDS codes with two information symbols, for which nonequivalent parity-check matrices are obtained. Simulation results show that for our constructed codes of short lengths, the automorphism group can significantly enhance the iterative decoding performance.