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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). 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.

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). 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 $\mathrm{ho}(M^G)_{\downarrow Y}(G^{\bullet+1}\otimes X,Y^{S^t}))$ for $t>0$. Here the source has the simplicial structure coming from a bar construction.

When the order of $G$ acts invertibly on these cohomology 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.

Of course, I am assuming some kind of nice enriched model structure here, so I am not answering your question exactly. On the other hand, in these case I think you can get away with less: You probably just need $Y$ to be a homotopy monoid in $\mathrm{Ho}(M)$ (to get the $E_2$ description as a bunch of abelian groups) and you only need the order of $G$ to act invertibly on this $E_2$ term in positive cohomological degrees.

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.

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). 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.

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). 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.

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). 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 $\mathrm{Ho}(M^G)$ lies 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.

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). 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.