  For groups of diffeomorphisms of $\T^2$ containing an Anosov diffeomorphism,
we give a complete classification for polycyclic Abelian-by-Cyclic group
actions on $\T^2$ up to both topological conjugacy and smooth conjugacy under
mild assumptions. Along the way, we also prove a Tits alternative type theorem
for some groups of diffeomorphisms of $\T^2$.
