There have been several proposed methods in the literature for securely distributing group keys and managing group dynamics for secure group communications. While these methods claim to be secure against passive adversaries, our focus has been on a more powerful adversary known as a strong active outsider adversary. This adversary has the ability to corrupt legitimate users, which can result in the leakage of crucial secret information to the adversary. Such information can enable the adversary to recover both current and past group keys. One commonly utilized approach for ensuring secure group communication is group key management schemes based on the Chinese remainder theorem (CRT). In this paper, we evaluate prominent CRT-based key management schemes in the presence of an active adversary. Our findings indicate that the adversary can exploit the leaked information of the corrupted user to break backward secrecy. As a result, we demonstrate that the CRT-based schemes found in the literature are insecure against strong active adversaries and are therefore unsuitable for practical applications.