Skew braces and the Galois correspondence for Hopf Galois structures

Let $L\mathrm{/}K$ be a Galois extension of fields with Galois group $\mathrm{\Gamma }$, and suppose $L\mathrm{/}K$ is also an $H$-Hopf Galois extension. Using the recently uncovered connection between Hopf Galois structures and skew left braces, we introduce a method to quantify the failure of surjectivity of the Galois correspondence from subHopf algebras of $H$ to intermediate subfields of $L\mathrm{/}K$, given by the Fundamental Theorem of Hopf Galois Theory. Suppose $L \otimes_K H = LN$where$N \cong (G, \star)$\mathrm{.}Thenthereexistsaskewleftbrace$(G, \star, \circ)$where$(G, \circ) \cong \Gamma$. We show that there is a bijective correspondence between intermediate fields $E$ between $K$ and $L$ and certain sub-skew left braces of $G$, which we call the $\circ$-stable subgroups of $(G,\star )$. Counting these subgroups and comparing that number with the number of subgroups of $\mathrm{\Gamma }\cong (G,\circ )$ describes how far the Galois correspondence for the $H$-Hopf Galois structure is from being surjective. The method is illustrated by a variety of examples.