Connected cubic graphs Gamma of twice odd order which admit an automorphism group acting semisytnmetrically are investigated. The structure of the automorphism group of Gamma modulo a subgroup which acts semiregularly on Gamma is determined. This identification is achieved by using the fundamental theorem of Goldschmidt [D.M. Goldschmidt. Automorphisms of trivalent graphs, Ann. of Math. (2) 111 (2) (1980) 377-406] and some small parts of the proof of the classification of the finite simple groups. (c) 2005 Elsevier Ltd. All rights reserved.