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

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.