Profinite groups whose closed subgroups are subnormal of defect at most two

In this paper, we classify profinite groups whose closed subgroups are subnormal of defect at most two, providing profinite versions of theorems due to Heineken and to Mahdavianary. We also establish that if G is a finitely generated periodic pro-π group, where π is a finite set of odd primes, and all its procyclic subgroups are subnormal of defect at most 2, then G is a finite π-group in which every abelian subgroup is subnormal of defect at most 2.