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.