Homotopy coherent action on differential graded vector space 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.
 A: Yes, you can always lift. (Assuming you work over a field, otherwise, I don't know.)
Note that an action of $G$ is the same as an action of the group algebra $k[G]$ where $k$ is your base field. You have a quadratic-linear (inhomogeneous) presentation of $k[G]$, with generators $g \in G$ and relations $g \cdot h = gh$. It's easy to check that this defines a Koszul algebra (the Koszul dual is the bar construction of $B(k[G])$).
Choosing representatives for homology classes, you can find a deformation retract of $C$ onto $H_*(C)$ as usual. You can then apply the Homotopy Transfer Theorem (see e.g. Theorem 47 of the paper Homotopy BV-algebras by Gálvez-Carrillo, Tonks, and Vallette for a statement in the context of inhomogeneous operads, which specializes to inhomogeneous algebras) to transfer the $k[G]$-action on $H_*(C)$ to an $\Omega(B(k[G]))$-action on $C$. Moreover $C$ and $H_*(C)$ are quasi-isomorphic seen as $\Omega(B(k[G]))$-modules. Since everything is concentrated in arity 1, you can view your operad as a non-symmetric one, so I don't think you need the characteristic zero assumption on $k$.
As for unicity, I am not sure. Yalin's paper Realization Spaces of Algebraic Structures on Cochains contains a way to compute the connected components of the space of all possible lifts of homotopy-coherent structures on $C$ given the action on $H_*(C)$. It should apply here.
