  Given a finite group $ G $, we study certain regular subgroups of the group
of permutations of $ G $, which occur in the classification theories of two
types of algebraic objects: skew left braces with multiplicative group
isomorphic to $ G $ and Hopf-Galois structures admitted by a Galois extension
of fields with Galois group isomorphic to $ G $. We study the questions of when
two such subgroups yield isomorphic skew left braces or Hopf-Galois structures
involving isomorphic Hopf algebras. In particular, we show that in some cases
the isomorphism class of the Hopf algebra giving a Hopf-Galois structure is
determined by the corresponding skew left brace. We investigate these questions
in the context of a variety of existing constructions in the literature. As an
application of our results we classify the isomorphically distinct Hopf
algebras that give Hopf-Galois structures on a Galois extension of degree $ pq
$ for $ p>q $ prime numbers.
