  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 $K \subseteq E \subseteq L$ 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, $L \otimes_K H
= LN$ where $N$ is a $G$-invariant regular subgroup of $\mathrm{Perm}(G)$, 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-Sz\'ep products of finite groups,
and in particular when $A^3 = 0$ or the Zappa-Sz\'ep 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 $\circ$ and $\star$ 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 $(G, \circ, \star)$ is a bi-skew brace of squarefree order
$2m$ where $(G, \circ) \cong Z_{2m}$ is cyclic and $(G, \star) = D_m$ is
dihedral, then for large $m$, $GC(Z_{2m},D_m), $ is close to 1/2 while $GC(D_m,
Z_{2m})$ is near 0.
