We consider problems pertaining to the navigation and localization of mobile mechanical systems such as robots or crafts. We focus on the recently introduced theory of two-frame systems, within the field of invariant Kalman filtering for state estimation of dynamical systems. We show that we can build a two-frame group (TFG) structure not only using rotations as its basic building block, as has been done so far, but also rotations and scalings. This then makes the TFG a generalization of the group of similarilty transformations Sim(d). This allows for new examples to fit into the theory of two-frame systems. Namely, we consider the localization of a car equipped with odometry based on wheels having unknown radius, and inertial navigation where the accelerometers are affected by an unknown scale factor.