$u$ a unitary element in $A\otimes H^0 \otimes H^0$. Then we say that $(\rho, u)$ is a *twisted coaction of $H^0$ on $A$ if (1) $(\rho\otimes{{\rm{id}}})\circ\rho={{\rm{Ad}}}(u)\circ({{\rm{id}}}\otimes\Delta^0 )\circ\rho$, (2) $(u\otimes 1^0 )({{\rm{id}}}\otimes\Delta^0 \otimes{{\rm{id}}})(u)=(\rho\otimes{{\rm{id}}}\otimes{{\rm{id}}})(u)
({{\rm{id}}}\otimes{{\rm{id}}}\otimes\Delta^0 )(u)$, (3) $({{\rm{id}}}\otimes h\otimes\epsilon^0 )(u)=({{\rm{id}}}\otimes\epsilon^0 \otimes h)(u)=\epsilon^0 (h)1$ for any $h\in H$.**

Let ${{\rm{Hom}}}(H, A)$ be the linear space of all linear maps from $H$ to $A$. Then by Sweedler [@Sweedler:Hopf pp67-70], it becomes a unital convolution $*$-algebra. Since $H$ is finite dimensional, ${{\rm{Hom}}}(H, A)$ is isomorphic to $A\otimes H^0$. For any element $x\in A\otimes H^0$, we denote by $\widehat{x}$ the element in ${{\rm{Hom}}}(H, A)$ induced by $x$. Similarly, we define ${{\rm{Hom}}}(H\times H, A)$. We identify $A\otimes H^0 \otimes H^0$ with ${{\rm{Hom}}}(H\times H, A)$. For any element $y\in A\otimes H^0 \otimes H^0$, we denote by $\widehat{y}$ the element in ${{\rm{Hom}}}(H\times H, A)$ induced by $y$. Furthermore, for a Hilbert $C^*$-bimodule $X$, let ${{\rm{Hom}}}(H, X)$ be the linear space of all linear maps from $H$ to $X$. We also identify ${{\rm{Hom}}}(H, X)$ with $X\otimes H^0$. For any element $x\in X\otimes H^0$, we denote by $\widehat{x}$ the element in ${{\rm{Hom}}}(H, X)$ induced by $x$.

For a twisted coaction $(\rho, u)$, we can consider the twisted action of $H$ on 