2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/102185A pseudo-Galois extension is shown to be a depth two extension. Studying its left bialgebroid, we construct an enveloping Hopf algebroid for the semi-direct product of groups, or more generally involutive Hopf algebras, and their module algebras. It is a type of cofibered sum of two inclusions of the Hopf algebra into the semi-direct product and its derived right crossed product. Van Oystaeyen and Panaite observe that this Hopf algebroid is non-trivially isomorphic to a Connes-Moscovici Hopf algebroid, which raises interesting comparative questions.16 pagesQuantum AlgebraRings and Algebras13B05, 16W30, 20L05, 81R50text