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Let $G$ be a nonabelian group. We show how a collection of compatibleendomorphisms $\psi_i:G\to G$ such that $\psi_i([G,G])\le Z(G)$ for all $i$allows us to construct a family of bi-skew braces called a brace block. Werelate this construction to other brace block constructions and interpret ourresults in terms of Hopf-Galois structures on Galois extensions. We givespecial consideration to the case where $G$ is of nilpotency class two, and weprovide several examples, including finding the maximal brace block containingthe group of quaternions.

Commutator-central maps, brace blocks, and {H}opf-{G}alois structures on {G}alois extensions