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A question about the smash product

Let be $H$ a finite dimensional semisimple-cosemisimple Hopf algebra, and $A$ be a finite dimensional $H$-module algebra. If $M$ is a finitely generated $A# H$-module.

Is $M$ a direct summand of $A # H\otimes_{A}(A # H #H^{*})\otimes_{A \# X$?,