  A 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.
