This paper proposes to use a newly-derived transformed inertial navigation system (INS) mechanization to fuse INS with other complementary navigation systems. Through formulating the attitude, velocity and position as one group state of group of double direct spatial isometries SE2(3), the transformed INS mechanization has proven to be group affine, which means that the corresponding vector error state model will be trajectory-independent. In order to make use of the transformed INS mechanization in inertial based integration, both the right and left vector error state models are derived. The INS/GPS and INS/Odometer integration are investigated as two representatives of inertial based integration. Some application aspects of the derived error state models in the two applications are presented, which include how to select the error state model, initialization of the SE2(3) based error state covariance and feedback correction corresponding to the error state definitions. Extensive Monte Carlo simulations and land vehicle experiments are conducted to evaluate the performance of the derived error state models. It is shown that the most striking superiority of using the derived error state models is their ability to handle the large initial attitude misalignments, which is just the result of log-linearity property of the derived error state models. Therefore, the derived error state models can be used in the so-called attitude alignment for the two applications. Moreover, the derived right error state-space model is also very preferred for long-endurance INS/Odometer integration due to the filtering consistency caused by its less dependence on the global state estimate.