Public Blockchain-based covert transmission (CT) can address the limitations of traditional CT methods. Monero is a blockchain-based cryptocurrency with strong privacy protection techniques. However, existing Monero-based CT methods are limited to unicast scenarios. If applied directly to group CT scenarios, they would lead to a significant increase in transaction volume as the number of receivers increases. Meanwhile, existing Bitcoin/Ethereum-based group CT methods at least face three challenges, including susceptibility to key and identity inference attacks, information leakage during off-chain negotiations, and exposure of communication channels.This paper proposes a Monero-Based Group CT approach (MBGCT), which enables on-chain group key (used for receivers to filter covert transactions and extract messages) issuance, fine-grained access control of messages, and covert transaction identification and decryption isolation. MBGCT can ensure confidentiality of group keys, unforgeability of messages, integrity of each transmitted message, obscurity of covert channels, isolation of key generation from key management, and enhanced anonymity. As a result, MBGCT can not only prevent information leakage and channel exposure, but also resist the attacks of entity impersonation, data tampering, key and identity inference. We implemented MBGCT in Monero client v0.18.1.0, and validated its capability of high embedding rates, low transaction fees, and high execution efficiency on the Monero public chain Stagenet.