The spectral sequence I constructed with Niles Johnson was precisely designed to handle questions of this sort (here is a version that is closer to the publication version: [T-algebra SS][1]). A special case of your question is considered in Section 5.1. Our methods require a suitably enriched model category (we focus on simplicial model categories), but it is easy to see that you get a similar spectral sequence for spectral or chain complex enrichments. In such a case the obstructions to the existence and uniqueness of a lift of a given map from $\mathrm{Ho}(M)^G$ to $\mathrm{Ho}(M^G)$ lie in the positive dimensional cohomology of $G$ with coefficients in various shifts of $M$.

When the order of $G$ is invertible in $M$, the transfer and restriction homomorphisms exhibit these groups as retracts of the cohomology of the *trivial* group with coefficients in the corresponding shifts of $M$. Of course these groups are trivial in positive degrees so there is a unique lift of each map.

I apologize that the linked reference is not quite complete. We are in the middle of making revisions for publication. The article will later appear in Advances in Mathematics. 

  [1]: http://www.nullplug.org/publications/obstruction-theory.pdf