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 $f\colon X\rightarrow Y$ from $\mathrm{Ho}(M)^G$ to $\mathrm{Ho}(M^G)$ lie in the positive dimensional Borel cohomology of $X$ with coefficients in various shifts of $Y$. Using a tensor and cotensor with spaces, we can write this more precisely: the obstructions are the $t$th cohomology groups of the cosimplicial abelian group $\mathrm{ho}(M^G)_{\downarrow Y}(G^{t+1}\otimes X,Y^{S^k}))$ for $t>0$ and $k=t$. Here the source has the simplicial structure coming from a bar construction.

When the order of $G$ is invertible in these mapping groups, the restriction and transfer homomorphisms exhibit these groups as retracts of the corresponding cohomology with the *trivial* group. 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