Suppose I have a group $G$ which acts on the homology $H_*(C)$ of a differential graded vector space $C$. Can I always lift this to a homotopy coherent $G$-action on $C$?

If not, are there any easy counterexamples?

What can be said about the uniqueness of such lifts?

Thanks for any hints.

Suppose I have a group $G$ which acts on the homology $H_*(C)$ of a differential graded vector space $C$. Can I always lift this to a homotopy coherent $G$-action on $C$?

My first naive thought was that $H_*(C)$ is homotopy equivalent to $C$, so I can transport the action on $H_*(C)$ to a homotopy coherent action on $C$. Will that give me a lift?

What can be said about the uniqueness of such lifts?

Thanks for any hints.