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In this second article, we continue to study classes of groups constructedfrom a functorial method due to Vaughan Jones. A key observation of the authorshows that these groups have remarkable diagrammatic properties that can beused to deduce their properties. Given any group and two of its endomorphisms,we construct a semidirect product. In our first article dedicated to thisconstruction, we classify up to isomorphism all these semidirect products whenone of the endomorphisms is trivial and described their automorphism group.  In this article we focus on the case where both endomorphisms areautomorphisms. The situation is rather different and we obtain semidirectproducts where the largest Richard Thompson's group $V$ is acting on somediscrete analogues of loop groups. Note that these semidirect products appearnaturally in recent constructions of quantum field theories. Moreover, theyhave been previously studied by Tanushevski and can be constructed via theframework of cloning systems of Witzel-Zaremsky. In particular, they provideexamples of groups with various finiteness properties and possiblecounterexamples of a conjecture of Lehnert on co-context-free groups.  We provide a partial classification of these semidirect products and describeexplicitly their automorphism group. Moreover, we prove that groups studied inthe first and second articles are never isomorphic to each other nor admit niceembeddings between them. We end the article with an appendix comparing Jonestechnology with Witzel-Zaremsky's cloning systems and with Tanushevski'sconstruction. As in the first article, all the results presented were possibleto achieve via a surprising rigidity phenomena on isomorphisms between thesegroups.

Classification of Thompson related groups arising from Jones technology II