We present a method to derive the complete spectrum of the lift Ga of a base digraph G, with voltage assignment a on a (finite) group G. The method is based on assigning to G a quotient-like matrix whose entries are elements of the group algebra C[G], which fully represents Ga. This allows us to derive the eigenvectors and eigenvalues of the lift in terms of those of the base digraph and the irreducible characters of G. Thus, our main theorem generalizes some previous results of Lov´asz and Babai concerning the spectra of Cayley digraphs

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