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In the quest in constructing conformal field theories (CFT) Jones hasdiscovered a beautiful and deep connection between CFT, Richard Thompson'sgroups and knot theory. This led to a powerful functorial framework forconstructing actions of particular groups arising from categories such asThompson's groups and braid groups. In particular, given a group and two of itsendomorphisms one can construct a semidirect product where the largestThompson's group $V$ is acting. These semidirect products have remarkablediagrammatic descriptions which were previously used to provide new examples ofgroups having the Haagerup property. They naturally appear in certain fieldtheories as being generated by local and global symmetries. Moreover, thesegroups occur in a construction of Tanushevski and can be realised usingBrin-Zappa-Szep's products with the technology of cloning systems ofWitzel-Zaremsky.  We consider in this article the class of groups obtained in that way whereone of the endomorphism is trivial leaving the case of two nontrivialendomorphisms to a second article. We provide an explicit description of allthese groups as permutational restricted twisted wreath products where $V$ isthe group acting and the twist depends on the endomorphism chosen. We classifythis class of groups up to isomorphisms and provide a thin description of theirautomorphism group thanks to an unexpected rigidity phenomena.

Classification of Thompson related groups arising from Jones technology I