here is a $C^*$-Hopf algebra automorphism $\lambda^0$ of $H^0$ such that the twisted coaction $(\rho, u)$ is strongly Morita equivalent to the twisted coaction $$(({{\rm{id}}}_B \otimes\lambda^0 )\circ\sigma \, , \, ({{\rm{id}}}_B \otimes\lambda^0 \otimes\lambda^0 )(v))$