The approach of instantiating authenticated group key exchange (GAKE) protocol from the multikey encapsulation mechanism (mKEM) has an important advantage of achieving classical requirement of GAKE security in one communication round. In spite of the limitations of this approach, for example, lack of forward secrecy, it is very useful in group environments when maximum communication efficiency is desirable. To enrich this mKEM-based GAKE construction, we suggest an efficient solution to convert this static GAKE framework into a partially dynamic scheme. Furthermore, to address the associated lack of forward-secrecy, we propose two variants of this generic construction which can also provide a means of forward secrecy at the cost of extra communication round. In addition, concerning associated implementation cost of deploying this generic GAKE construction in elliptic curve cryptosystem, we compare the possible instantiations of this model from existing mKEM algorithms in terms of the number of elliptic curve scalar multiplications.