The commutator Hopf Galois extensions

Let $H$ be a finite dimentional Hopf algebra over a field $k$ and $H^*$ the dual Hopf algebra of $H$. Then a commutator right $H^*$-Galois extension $B$ of $B^H$ is characterized in terms of the smash product $B\#H$ and some relationships between such a $B$ and the Hopf Galois Azumaya or Hopf Galois Hirata extensions are also given.