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

Skew left braces and isomorphism problems for Hopf-Galois structures on Galois extensions