Identifying subgroups with differential responses to treatment is pivotal in randomized clinical trials, as tailoring treatments to specific subgroups can advance personalized medicine. Upon trial completion, identifying best-performing subgroups–those with the most beneficial treatment effects–is crucial for optimizing resource allocation or mitigating adverse treatment effects. However, traditional clinical trials are not customized for the goal of identifying best-performing subgroups because they typically pre-define subgroups at the beginning of the trial and adhere to a fixed subgroup treatment allocation rule, leading to inefficient use of experimental efforts. While some adaptive experimental strategies exist for the identification of the single best subgroup, they commonly do not enable the identification of the best set of subgroups. To address these challenges, we propose a dynamic subgroup identification covariate-adjusted response-adaptive randomization (CARA) design strategy with the following key features: (i) Our approach is an adaptive experimental strategy that allows the dynamic identification of the best subgroups and the revision of treatment allocation towards the goal of correctly identifying the best subgroups based on collected experimental data. (ii) Our design handles ties between subgroups effectively, merging those with similar treatment effects to maximize experimental efficiency. In the theoretical investigations, we demonstrate that our design has a higher probability of correctly identifying the best set of subgroups compared to conventional designs. Additionally, we prove the statistical validity of our estimator for the best subgroup treatment effect, demonstrating its asymptotic normality and semiparametric efficiency. Finally, we validate our design using synthetic data from a clinical trial on cirrhosis.