A two-stage enrichment design is a type of adaptive design, which extends a stratified design with a futility analysis on the marker negative cohort at the first stage, and the second stage can be either a targeted design with only the marker positive stratum, or still the stratified design with both marker strata, depending on the result of the interim futility analysis. In this paper we consider the situation where the marker assay and the classification rule are possibly subject to error. We derive the sequential tests for the global hypothesis as well as the component tests for the overall cohort and the marker-positive cohort. We discuss the power analysis with the control of the type-I error rate and show the adverse impact of the misclassification on the powers. We also show the enhanced power of the two-stage enrichment over the one-stage design, and illustrate with examples of the recent successful development of immunotherapy in non-small-cell lung cancer.