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Let $L/K$ be a Galois extension of fields with Galois group $\Gamma$, andsuppose $L/K$ is also an $H$-Hopf Galois extension. Using the recentlyuncovered connection between Hopf Galois structures and skew left braces, weintroduce a method to quantify the failure of surjectivity of the Galoiscorrespondence from subHopf algebras of $H$ to intermediate subfields of $L/K$,given by the Fundamental Theorem of Hopf Galois Theory. Suppose $L \otimes_K H= LN$ where $N \cong (G, \star)$. Then there exists a skew left brace $(G,\star, \circ)$ where $(G, \circ) \cong \Gamma$. We show that there is abijective correspondence between intermediate fields $E$ between $K$ and $L$and certain sub-skew left braces of $G$, which we call the $\circ$-stablesubgroups of $(G, \star)$. Counting these subgroups and comparing that numberwith the number of subgroups of $\Gamma \cong (G, \circ)$ describes how far theGalois correspondence for the $H$-Hopf Galois structure is from beingsurjective. The method is illustrated by a variety of examples.

Skew braces and the Galois correspondence for Hopf Galois structures