twisted Dijkgraaf-Witten theories (Definition \[DW-theories\]).

Using the pushforward construction from [@extofk], we endow the equivariant Dijkgraaf-Witten models from [@maiernikolausschweigerteq] with a twist, thereby obtaining a new class of examples of extended homotopy quantum field theories (Definition \[defgendwtheory\]) and, by evaluation of those on the circle, a new class of equivariant modular tensor categories (Theorem \[Theorem: Equivariant category on S1\]).

Acknowledgements {#acknowledgements .unnumbered}
----------------

We would like to thank Christoph Schweigert and Richard Szabo for the constant support of this project and helpful discussions. Furthermore, we are grateful to Severin Bunk for useful comments on a draft version of this paper.

LM is supported by the Doctoral Training Grant ST/N509099/1 from the UK Science and Technology Facilities Council (STFC). LW is supported by the RTG 1670 “Mathematics inspired by String theory and Quantum Field Theory”.

Extended homotopy quantum field theories\[secexthqft\]
======================================================

For the notion of a homotopy quantum field theory from [@turaevhqft] there exists an extended version. The full definition is given in [@extofk] and will be recalled below.

The domain of definition for an extended homotopy quantum field theory with target space $T$ is the symmetric monoidal bordism bicategory $T\text{-}{{\mathbf{Cob}}}(n,n-1,n-2)$ of $T$-bordisms.

In order to define $T\text{-}{{\mathbf{Cob}}}(n,n-1,n-2)$, the notion of 