Jones' technology, developed by Vaughan Jones during his exploration of the connections between conformal field theory and subfactors, is a powerful mechanism for generating actions of groups coming from categories, notably Richard Thompson's groups F, T, and V.

We give a structure theorem for the isomorphisms between a class of split extensions of Thompson's group V arising from Jones' technology, generalising results of Brothier. Using this structure theorem, we classify a family of unrestricted, twisted, permutational wreath products up to isomorphism. We also decompose their automorphism groups in the untwisted case. This complements results of Brothier for the corresponding restricted wreath products, and extends classical results.

These unrestricted wreath products arise from applying Jones' technology to contravariant functors. In contrast, the current literature focuses on groups constructed using covariant functors, appearing in work of Brothier, Tanushevski, and Witzel-Zaremsky. Our point of view unifies these constructions, and highlights the duality between them.

I thank Arnaud Brothier for the many meetings and invaluable feedback on drafts. I also thank Ian {Doust} for his writing course and helpful comments on an early version of this article. I am grateful for useful exchanges with Ryan Seelig and Dilshan Wijesena, and for Renee Lim's proofreading. Most diagrams were made using TikZit, and I thank the contributors for developing this incredible software.