2 In original form, the question admitted a trivial answer if "nontrivial outer automorphism" is read usual as "non-inner automorphism". Really I am interested in "non-obvious" ones, so ask for the outer automorphism group.

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the permutation of $\mathbb{Z}$ which interchanges $r_1+km_1$ and $r_2+km_2$ for every $k \in \mathbb{Z}$ and which fixes everything else.

The group ${\rm CT}(\mathbb{Z})$ generated by all class transpositions of $\mathbb{Z}$ is simple (cf. Math. Z. 264 (2010), no. 4, 927-938, http://dx.doi.org/10.1007/s00209-009-0497-8).

Does

Find the outer automorphism group of ${\rm CT}(\mathbb{Z})$have nontrivial outer automorphisms?.

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# Outer automorphisms of an infinite simple group

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the permutation of $\mathbb{Z}$ which interchanges $r_1+km_1$ and $r_2+km_2$ for every $k \in \mathbb{Z}$ and which fixes everything else.

The group ${\rm CT}(\mathbb{Z})$ generated by all class transpositions of $\mathbb{Z}$ is simple (cf. Math. Z. 264 (2010), no. 4, 927-938, http://dx.doi.org/10.1007/s00209-009-0497-8).

Does the group ${\rm CT}(\mathbb{Z})$ have nontrivial outer automorphisms?