  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 $F \subseteq T \subseteq V$.
  We give a structure theorem for the isomorphisms between 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, and decompose their
automorphism groups in the untwisted case.
  These unrestricted wreath products arise from applying Jones' technology to
contravariant monoidal functors. In contrast, using covariant functors,
Brothier constructed a large class of restricted wreath products, classified
them up to isomorphism, and completely described their automorphism groups. Our
work broadens Brothier's findings and highlights the duality between groups
constructed using covariant and contravariant functors.
