@Ivan and Nick: Indeed. -- Given that there are open problems which on a first glance look too elementary even to be posed as student questions, I would be very careful to attribute a question as "not research-level" unless I know the answer.

By the way -- I asked this question already in 2010 as Problem 17.59 in the Kourovka Notebook.

I don't see how this should help -- but maybe you have some good idea? One also has a continuous action of ${\rm CT}(\mathbb{Z})$ on $\mathbb{Z}$, endowed with a topology by taking the set of all residue classes as a basis -- but I don't see so far that this helps further. But maybe you have better luck with your approach. By the way, I prefer the notation ${\rm RCWA}(\mathbb{Z})$ for the group of all residue-class-wise affine permutations (that's the notation I used in my publications so far).

-- And, besides automorphisms not coming from inner automorphisms of ${\rm Sym}(\mathbb{Z})$, the latter would be particularly interesting.

Thanks. - If the answer to Question <mathoverflow.net/questions/112469> is negative, this would yield further spatial outer automorphisms

But a little modification turns it into one: $n \mapsto -n-1$ is indeed an automorphism.

No, $n \mapsto -n$ is not an automorphism. The group ${\rm CT}(\mathbb{Z})$ stabilizes $\mathbb{N}_0$ setwise, but for example the conjugate of its element $\tau_{0(2),1(2)}$ under the mapping $n \mapsto -n$ doesn't (it moves 0 to -1).