Commuting Hopf-Galois Structures on a Separable Extension

Let $ L/K $ be a finite separable extension of local or global fields in any characteristic, let $ H_{1}, H_{2} $ be two Hopf algebras giving Hopf-Galois structures on the extension, and suppose that the actions of $ H_{1}, H_{2} $ on $ L $ commute. We show that a fractional ideal $ {\mathfrak B} $ of $ L $ is free over its associated order in $ H_{1} $ if and only if it is free over its associated order in $ H_{2} $. We also study which properties these associated orders share.