Proper validation of a hypothesis requires adherence to appropriate assumptions about the data and model under consideration. It is interesting to see if a particular hypothesis test is robust to deviations from such assumptions. These topics have been extensively studied in classical parametric hypothesis testing. This work then considers such questions in randomized tests. Specifically, are these nonparametric tests invariant or robust to breaking assumptions? General randomized tests are considered in this work. It randomizes the data through the application of group actions from well-chosen compact topological groups with respect to the Haar scale. Inferences made using group behavior have been shown to be consistent with standard distributional approaches. It has also been shown that robustness is often asymptotically achievable even when the data do not necessarily satisfy the invariance assumption. An example would be a particular hypothesis test. These are the one-sample position test and the reflection group, the two-sample test for mean equivalence and permutation symmetry groups, and the Durbin-Watson test for special orthogonal groups of serial correlation and n-dimensional rotation.