On the Galois correspondence for Hopf Galois structures arising from finite radical algebras and Zappa-Szép products

Let $L/K$ be a G-Galois extension of fields with an H-Hopf Galois structure of type N. We study the ratio $GC(G, N)$, which is the number of intermediate fields E with K ⊆ E ⊆ L that are in the image of the Galois correspondence for the H-Hopf Galois structure on $L/K$, divided by the number of intermediate fields. By Galois descent, L ⊗K H = LN where N is a G-invariant regular subgroup of Perm(G), and then $GC(G, N)$ is the number of G-invariant subgroups of N, divided by the number of subgroups of G. We look at the Galois correspondence ratio for a Hopf Galois structure by translating the problem into counting certain subgroups of the corresponding skew brace. We look at skew braces arising from finite radical algebras A and from Zappa-Szép products of finite groups, and in particular when A³ = 0 or the Zappa-Szép product is a semidirect product, in which cases the corresponding skew brace is a bi-skew brace, that is, a set G with two group operations ∘ and in such a way that G is a skew brace with either group structure acting as the additive group of the skew brace. We obtain the Galois correspondence ratio for several examples. In particular, if (G, ∘, ) is a bi-skew brace of squarefree order $2m$ where (G, ∘) Z₂ₘ is cyclic and (G, ) = Dₘ is dihedral, then for large m, GC(Z₂ₘ,Dₘ), is close to 1/2 while GC(Dₘ, Z₂ₘ) is near 0.