Extending a group to a ring defining a new operator using group automorphisms.

Question

Let $\mathcal{G} = \langle A, + \rangle$ be an infinite group. I was wondering whether it is possible to extend $\mathcal{G}$ to a commutative ring $\mathcal{R}= \langle A, + , * \rangle$ by associating to each element $a \in A$ an automorphism $\phi_a$ on $\mathcal{G}$ in such a way that the product * will be defined as: $a*x=y \underset{def}{\Leftrightarrow} \phi_a(x)=y$. If the answer is positive, it would be great to sketch the procedure for the definition of the product $*$.