  Let $\phi$ be an automorphism of a free group $F_n$ of rank $n$, and let
$M_{\phi}=F_n \rtimes_{\phi} \mathbb{Z}$ be the corresponding mapping torus of
$\phi$. We study the group $Out(M_{\phi})$ under certain technical conditions
on $\phi$. Moreover, in the case of rank 2, we classify the cases when this
group is finite or virtually cyclic, depending on the conjugacy class of the
image of $\phi$ in $GL_2(\mathbb{Z})$.
