Suppose $G$ is a finite group which acts on a $C^\*-$algebra which we denote by $A$. I was wondering if there is a naturally induced coaction on $A\otimes C(G)$, here $C(G)$ denotes functions on $G$.
I thank you for the attention and the help.



Coaction of a group
The way I would write this is as follows and I would appreciate users calling to attention any errors.


Take a linear action $\Phi:A\times G\rightarrow A$ and extend it to a bilinear map 
$$\Phi\_1:A\times \mathbb{C}G\rightarrow A.$$
Now using the universal property we have a linear map
$$\tilde{\Phi}:A\otimes \mathbb{C}G\rightarrow A.$$
Now take the dual of this map:
$$\tilde{\Phi}^\*:A^\*\rightarrow(A\otimes \mathbb{C}G)^\*\cong A^\*\otimes \mathbb{C}G^\*\cong A^\*\otimes C(G),$$
and $\tilde{\Phi}^\*$ is a coaction on $A^\*$.


I *think* you can also do the following to induce a coaction on $A$ rather than its dual. If the linear action is given by a matrix
$$\Phi(a,g)=\rho(g^{-1})(a),$$
for a $\rho: G\rightarrow \hom(A)$, then we have that
$$\kappa\_\Phi(a)=\sum\_{g\in G}\rho(g)a\otimes\mathbb{1}\_{\{g\}},$$
is a coaction $\kappa\_{\Phi}:A\to A\otimes C(G)$. I do believe this works but am open to correction on that. Is it as natural as $\tilde{\Phi}^\*$?


