Let $K$ be an algebraically closed field and $H$ a finite dimensional semisimple-cosemisimple Hopf $K$-algebra, and let $A$ be a finite dimensional $H$-module algebra and $H^{\ast}= Hom_{K}(H, K)$.

Is $A$ a direct summand of $A \# H \# H^\ast$ as $A$-bimodules?