Automorphism of $\mathbb{Z}\rtimes\mathbb{Z}$ I'm looking for a description of  $\mathrm{Aut}(\mathbb{Z}\rtimes\mathbb{Z})$.
I've tried an unsuccessfully combinatorical approac, does anymore have some hints?
Thank you.

Write the group as $G=\langle x,t\mid txt^{-1}=x^{-1}\rangle$. 
Exercise:
the automorphism group is generated by the automorphisms $u,s,z$, where $$u(x)=x,\; u(t)=tx,\quad s(x)=x^{-1},\; s(t)=t, \quad z(x)=x,\; z(t)=t^{-1};$$ check that it is isomorphic to the direct product of an infinite dihedral group (generated by $u,s$) and the cyclic group of order two $\langle z\rangle$. Deduce that the outer automorphism group is a direct product of two cyclic groups of order two, generated by the representatives of $u$ and $z$.
Hint: first check that $\langle x\rangle$ is a characteristic subgroup of $G$ (warning: it is not the derived subgroup).
