- This paper concerns the structure of the group of local unitary cocycles, also called the gauge group, of an E 0 -semigroup. The gauge group of a spatial E 0 -semigroup has a natural action on the set of units by operator multiplication. Arveson has characterized completely the gauge group of E 0 -semigroups of type I, and as a consequence it is known that in this case the gauge group action is transitive. In fact, if the semigroup has index k , then the gauge group action is transitive on the set of ( k + 1 ) -tuples of appropriately normalized independent units. An action of the gauge group having this property is called ( k + 1 ) -fold transitive. We construct examples of E 0 -semigroups of type II and index 1 which are not 2-fold transitive. These new examples also illustrate that an E 0 -semigroup of type II k need not be a tensor product of an E 0 -semigroup of type II 0 and another of type I k .