We denote its comultiplication, counit and antipode by $\Delta^0$, $\epsilon^0$ and $S^0$, respectively. There is the distinguished projection $e$ in $H$. We note that $e$ is the Haar trace on $H^0$. Also, there is the distinguished projection $\tau$ in $H^0$ which is the Haar trace on $H$. Since $H$ is finite dimensional, $H\cong\oplus_{k=1}^L M_{f_k}({\mathbf C})$ and $H^0 \cong \oplus_{k=1}^K M_{d_k }({\mathbf C})$ as $C^*$-algebras. Let $\{v_{ij}^k \, | \, k=1,2, \dots L , \, i, j=1,2,\dots, f_k \}$ be a system of matrix units of $H$. Let $\{w_{ij}^k \, | \, k=1,2,\dots,K ,\, i, j=1,2, \dots , d_k \}$ be a basis of $H$ satisfying Szymański and Peligrad’s [@SP:saturated Theorem 2.2,2], which is called a system of *comatrix units of $H$, that is, the dual basis of the system of matrix units of $H^0$. Also, let $\{\phi_{ij}^k \, | \, k=1,2,\dots,K, \, i,j=1,2,\dots,d_k \}$ and $\{\omega_{ij}^k \, | \, k=1,2,\dots, L, \, i, j=1,2,\dots,f_k \}$ be systems of matrix units and comatrix units of $H^0$, respectively. Let $A$ be a unital $C^*$-algebra. We recall the definition of a twisted coaction $(\rho, u)$ of $H^0$ on $A$ (See [@KT1:inclusion], [@KT2:coaction]). Let $\rho$ be a weak coaction of $H^0$ on $A$ and 