  In the quest in constructing conformal field theories (CFT) Jones has
discovered a beautiful and deep connection between CFT, Richard Thompson's
groups and knot theory. This led to a powerful functorial framework for
constructing actions of particular groups arising from categories such as
Thompson's groups and braid groups. In particular, given a group and two of its
endomorphisms one can construct a semidirect product where the largest
Thompson's group $V$ is acting. These semidirect products have remarkable
diagrammatic descriptions which were previously used to provide new examples of
groups having the Haagerup property. They naturally appear in certain field
theories as being generated by local and global symmetries. Moreover, these
groups occur in a construction of Tanushevski and can be realised using
Brin-Zappa-Szep's products with the technology of cloning systems of
Witzel-Zaremsky.
  We consider in this article the class of groups obtained in that way where
one of the endomorphism is trivial leaving the case of two nontrivial
endomorphisms to a second article. We provide an explicit description of all
these groups as permutational restricted twisted wreath products where $V$ is
the group acting and the twist depends on the endomorphism chosen. We classify
this class of groups up to isomorphisms and provide a thin description of their
automorphism group thanks to an unexpected rigidity phenomena.
