Push-forward of Hopf--Galois extensions: the non central case

We study the push-forward of Hopf–Galois extensions as the algebraic counterpart of the pullback of principal bundles. We apply the theory of twisted tensor product algebras to endow covariant extensions of modules along a map $\mathsf{F}$ with an algebra structure, under compatibility conditions between $\mathsf{F}$ and the twisting map. The push-forward of an $H$-Galois extension $B\subset A$ along a map $\mathsf{F}:B\to C$ is an $H$-Galois extension of $C$. The corresponding Ehresmann–Schauenburg algebroids are compared.