G-equivariant Alterfold TQFT in any dimension

For a discrete group G, we construct an $(n+1)$-dimensional G-equivariant Alterfold topological quantum field theory from a G-equivariant Sⁿ-algebra satisfying strong semisimplicity, complete finiteness, and Morita invariance. We use state sum on triangulation to define a partition function and prove that it is invariant under local surgery relations and is invariant under gauge transformations relative to the boundary. If the input is given a reflection-positive structure, the theory is unitary. For $n=2$ and input reconstructed from a strict faithful spherical G-fusion category, the restriction to ordinary G-cobordisms is naturally isomorphic to the Turaev--Virelizier state-sum HQFT {turaev20123}. A Tambara-Yamagami example recovers the Quon language {liu2017quon} as a Z₂-equivariant Alterfold TQFT.