The automorphism group of a free-by-cyclic group in rank 2

Let ¡ be an automorphism of a free group F 2 of rank 2 and let M ¡ = F 2 o ¡ Z be the corresponding mapping torus of ¡ . We prove that the group Out ( M ¡ ) is usually virtually cyclic. More- over, we classify the cases when this group is Ønite depending on the conjugacy class of the image of ¡ in GL 2 ( Z )Award-winnin