We propose a method to estimate the average fidelity using the unitary 2t-design of a twirled noisy channel, which is suitable for large-scale quantum circuits. Compared with the unitary 2-design in randomized benchmarking, the unitary 2t-design for the twirling of noisy channels is more flexible in construction and can provide more information. In addition, we prove that the proposed method provides an efficient and reliable estimation of the average fidelity in benchmarking multistage quantum gates and estimating the weakly gate- and time-dependent noise. For time-dependent noise, we provide a scheme of moment superoperator to analyze the noise in different experiments. In particular, we give a lower bound on the average fidelity of a channel with imperfect implementation of benchmarking and state preparation and measurement errors (SPAM).