  Let $L/K$ be any finite Galois extension with Galois group $G$. It is known
by Chase and Sweedler that the Hopf--Galois correspondence is injective for
every Hopf--Galois structure on $L/K$, but it need not be bijective in general.
Hopf--Galois structures are known to be related to skew braces, and recently,
the first-named author and Trappeniers proposed a new version of this
connection with the property that the intermediate fields of $L/K$ in the image
of the Hopf--Galois correspondence are in bijection with the left ideals of the
associated skew brace. As an application, they classified the groups $G$ for
which the Hopf--Galois correspondence is bijective for every Hopf--Galois
structure on any $G$-Galois extension. In this paper, using a similar approach,
we shall classify the groups $N$ for which the Hopf--Galois correspondence is
bijective for every Hopf--Galois structure of type $N$ on any Galois extension.
