  Let $G$ be a finite nonabelian group, and let $\psi:G\to G$ be a homomorphism
with abelian image. We show how $\psi$ gives rise to two Hopf-Galois structures
on a Galois extension $L/K$ with Galois group (isomorphic to) $G$; one of these
structures generalizes the construction given by a ``fixed point free abelian
endomorphism'' introduced by Childs in 2013. We construct the skew left brace
corresponding to each of the two Hopf-Galois structures above. We will show
that one of the skew left braces is in fact a bi-skew brace, allowing us to
obtain four set-theoretic solutions to the Yang-Baxter equation as well as a
pair of Hopf-Galois structures on a (potentially) different finite Galois
extension.
