Groups of automatic automorphisms of some automatic structures

We prove that the group $\operatorname{Aut}_a(\{0,1\}^\ast)$ of all automatic automorphisms of the regular set $\{0,1\}^\ast$ and the group $\operatorname{Aut}_a(\mathbb{Q})$ of all automatic automorphisms of the automatic model $\mathbb{Q}=(\{0,1\}^\ast,\preccurlyeq_{lex})$ have undecidable theories, which implies that they have no automatic presentations.