A number of different ion species are known to exist in the magnetotail. In particular, cold lobe ions and current-carrying hot plasma sheet ions are a permanent feature of the magnetotail and result in various types of instabilities. One such instability is the ion-ion kink mode. Detailed properties of this mode in the magnetotail are investigated in a two series paper using a combination of linear Vlasov theory, three-dimensional (3-D) full particle and hybrid (fluid electron, kinetic ions) simulations. Here we consider the linear properties of the mode in detail. Although the mode shows similarities to a velocity driven instability, its linear mode properties do exhibit dependencies on the kinetic details of the secondary ion population.