Of course, it is highly desirable to be able to compute the amplitude by cutting ${\Sigmait}$ into simpler pieces. This locality principle for the holonomy of flat gerbes was made precise in [@turaevhqft I.2.1] by the statement that a cocycle $\theta \in Z^n(T;{\operatorname{U}}(1))$ gives rise to a homotopy quantum field theory with target space $T$, a so-called *primitive homotopy quantum field theory*. The main goal of this paper is to make this locality statement precise one categorical level higher and to construct from $\theta$ an *extended* homotopy quantum field theory with target space $T$ in the sense of [@extofk]. Thereby we get a very accessible class of examples for extended homotopy quantum field theories and, by specifying $T$ to be the classifying space $BG$ of a finite group $G$, for extended $G$-equivariant topological field theories. These theories are invertible (Definition \[defhqftext\]).

The flatness assumption throughout the article ensures that gerbes and their parallel transport can be described in purely topological terms. A description of non-flat gerbes is much more involved due to smoothness issues. The interpretation of the parallel transport of gerbes in the non-flat case in terms of homotopy quantum field theory has been addressed in the two-dimensional non-extended 