This paper proposes a state estimation methodology for invariant systems on Lie groups where outputs of the system are measured with delay. The proposed method is based on cascading an observer and a predictor. The observer uses delayed measurements and provides estimates of delayed states. The predictor uses those estimates together with the current inputs of the system to compensate for the delay and to provide a prediction of the current state of the system. We consider three classes of left invariant, right-invariant, and mixed-invariant systems and propose predictors tailored to each class. The key contribution of the paper is to exploit the underlying symmetries of systems to design novel predictors that are computationally simple and generic, in the sense that they can be combined with any stable observer or filter. We provide a rigorous stability analysis demonstrating that the prediction of the current state converges to the current system trajectory if the observer state converges to the delayed system trajectory. The good performance of the proposed approach is demonstrated using a sophisticated Software-In-The-Loop simulator indicating the robustness of the observer predictor methodology even when large measurement delays are present. (C) 2016 Elsevier Ltd. All rights reserved.