This paper proposes two one-round authenticated group key exchange protocols from newly employed cryptographic invariant maps (CIMs): one is secure in the quantum random oracle model and the other resists against maximum exposure where a non-trivial combination of secret keys is revealed. The security of the former (resp. latter) is proved under the n-way decisional (resp. n-way gap) Diffie–Hellman assumption on the CIMs in the quantum random (resp. random) oracle model. We instantiate the proposed protocols on the hard homogeneous spaces with limitation where the number of the user group is two. In particular, the protocols instantiated by using the CSIDH, commutative supersingular isogeny Diffie–Hellman, key exchange are currently more realistic than the general n-party CIM-based ones due to its realizability. Our two-party one-round protocols are secure against quantum adversaries.