Automorphism group of $\Bbb Z\times \Bbb Z \times \Bbb Z$ is $SL(3,\Bbb Z)$. I am wondering what the automorphism group of $(\Bbb Z\times \Bbb Z) \rtimes \Bbb Z$ would be. Generator c of $\Bbb Z$ is $c \in Aut(\Bbb Z \times \Bbb Z)=SL(2,Z)$. So c can be taken to be either of the two generators of $SL(2,Z)$.

This is the mapping class group calculation of a torus bundle over a circle with monodromy $c$.