@user42024 Yes that is what I am trying to compute. I'll see if can bring the topology of $ X$.

$M^*$ shows up as the cohomology of a space $X$ with an action of $G$. Probably I should rephrase the problem, as this computation is the group cohomology of $G$ with coefficients in the cochains of $X$ (or the $G$-equivariant cohomology of $X$).

Thanks!, your answer completely clarify my question. However, I am interested in spaces that admit a "kaehlerian cohomology class" (your third bullet item) but are not Kaehler manifolds. I thought that projective complex varieties do the job but they are Kaehler as your answer suggest. Do you happen to know an example of what I am looking for? Thanks again!!

Thanks! this add a lot of things for me to check out; why is necessary the assumption on considering coefficients over a field here?

Would you mind to illustrate a little bit why the action becomes what you say? it isn't that clear for me