The automorphism group of a free-bycyclic group in rank 2

Let φ be an automorphism of a free group F2 of rank 2 and let Mφ = F2 oφ Z be the corresponding mapping torus of φ. We prove that the group Out(Mφ) is usually virtually cyclic. Moreover, we classify the cases when this group is finite depending on the conjugacy class of the image of φ in GL2(Z).