For a (unital) \(C^*\) -algebra \(\mathcal{A}\) , we construct a \(C^*\) -algebraic discrete quantum group (DQG) \(\mathcal{Q}_\textrm{aut}(\mathcal{A})\) , coacting on \(\mathcal{A}\) , which is a quantum generalization of \({\text{ A }ut}(\mathcal{A})\) in the framework of discrete quantum groups, in the sense that any other coaction of a DQG on \(\mathcal{A}\) factors through the above coaction of \(\mathcal{Q}_\textrm{aut}(\mathcal{A})\) . We prove by an explicit calculation that if any Kac-type \(C^*\) -algebraic discrete quantum group \(\mathcal {Q}\) has a ‘weakly faithful’ coaction on \(C(S^1)\) which is ‘linear’ in the sense that it leaves the space spanned by \(\{ Z, \overline{Z} \}\) invariant, then \(\mathcal {Q}\) must be classical i.e. isomorphic with \(C_0(\Gamma )\) for some discrete group \(\Gamma \) . This parallels the well-known result of non-existence of genuine compact quantum group symmetry obtained by the first author and his collaborators Goswami (Quantum isometry groups, Infosys Science Foundation Series, Springer, Cham, 2016) and the references therein).
