“…Depth two theory is a type of Galois theory for noncommutative ring extensions, where the Galois group in field theory is replaced by a Hopf algebroid (with or perhaps without antipode). The Galois theory of depth two ring extensions has been studied in a series of papers by the author [14,15,16,17] in collaboration with Nikshych [12,11], Szlachányi [13], and Külshammer [10], with a textbook treatment by Brzeziński and Wisbauer [2]. There are a number of issues that remain unexplored or unanswered in full including chirality [14,15], normality [10,16], a Galois inverse problem and a Galois correspondence problem [25].…”