In this paper the automorphism groups of connected cubic Cayley graphs of order 2pq for distinct odd primes p and q are determined. As an application, all connected cubic non-symmetric Cayley graphs of order 2pq are classified and this, together with classifications of connected cubic symmetric graphs and vertex-transitive non-Cayley graphs of order 2pq given by the last two authors, completes a classification of connected cubic vertex-transitive graphs of order 2pq.
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