I'm not sure if this is exactly what you were asking about, but one way of parsing your question is to ask when a homomorphism from $G$ to $\operatorname{MCG}(M)$ is induced from a homomorphism $G \to \operatorname{Diff}(M)$ (or replace Diff with your favorite alternative). This is a very general (and very hard!) question, even in the case of $M$ a surface, where it goes by the name Nielsen realization problem. There's a lot of literature on this, so I'll give you some of the highlights.

The term "Nielsen realization problem" comes from the work of Nielsen in the 20's, who showed that if $G$ is a finite cyclic group, every homomorphism of $G$ into $\operatorname{Mod}(\Sigma_g)$ can be promoted into an action by homeomorphisms (indeed, all the way up to a group of isometries with respect to a hyperbolic metric; equivalently a group of automorphisms of some Riemann surface). In the '80s, Kerckhoff showed that the same result holds when $G$ is any finite group.

In the opposite direction, Morita showed that the full mapping class group of a surface cannot act by diffeomorphisms on that surface. There's now zillions of proofs of this, but one nice thing about Morita's argument is that he produces cohomological obstructions for a subgroup of $\operatorname{Mod}(\Sigma_g)$ to lift to $\operatorname{Diff}(\Sigma_g)$. Precisely, he shows that the MMM classes $e_i$ for $i \ge 3$ obstruct such a lift, in the sense that if $e_i \ne 0$, no such lift exists.

This is a beautiful thread of ideas, but it doesn't come close to being a complete answer; in particular this doesn't say anything in the case that $G$ has cohomological dimension $5$ or lower. There has been some work dedicated to approaching this problem from a more dynamical point of view; see especially the work of Franks and Handel giving a proof of Morita's theorem using dynamics.

In general, this problem is really hard! We don't even know if every homomorphism from a surface group (i.e. $G \cong \pi_1(\Sigma_h)$) into $\operatorname{Mod}(\Sigma_g)$ admits a lift to $\operatorname{Diff}(\Sigma_g)$...

I'm not sure if this is exactly what you were asking about, but one way of parsing your question is to ask when a homomorphism from $G$ to $\operatorname{MCG}(M)$ is induced from a homomorphism $G \to \operatorname{Diff}(M)$ (or replace Diff with your favorite alternative). This is a very general (and very hard!) question, even in the case of $M$ a surface, where it goes by the name Nielsen realization problem. There's a lot of literature on this, so I'll give you some of the highlights.

The term "Nielsen realization problem" comes from the work of Nielsen in the 20's, who showed that if $G$ is a finite cyclic group, every homomorphism of $G$ into $\operatorname{Mod}(\Sigma_g)$ can be promoted into an action by homeomorphisms (indeed, all the way up to a group of isometries with respect to a hyperbolic metric; equivalently a group of automorphisms of some Riemann surface). In the '80s, Kerckhoff showed that the same result holds when $G$ is any finite group.

In the opposite direction, Morita showed that the full mapping class group of $\Sigma_g$ cannot lift to $\operatorname{Diff}(\Sigma_g)$. There's now zillions of proofs of this, but one nice thing about Morita's argument is that he produces cohomological obstructions for a subgroup of $\operatorname{Mod}(\Sigma_g)$ to lift to $\operatorname{Diff}(\Sigma_g)$. Precisely, he shows that the MMM classes $e_i$ for $i \ge 3$ obstruct such a lift, in the sense that if $e_i \ne 0$, no such lift exists.

This is a beautiful thread of ideas, but it doesn't come close to being a complete answer; in particular this doesn't say anything in the case that $G$ has cohomological dimension $5$ or lower. There has been some work dedicated to approaching this problem from a more dynamical point of view; see especially the work of Franks and Handel giving a proof of Morita's theorem using dynamics.

In general, this problem is really hard! We don't even know if every homomorphism from a surface group (i.e. $G \cong \pi_1(\Sigma_h)$) into $\operatorname{Mod}(\Sigma_g)$ admits a lift to $\operatorname{Diff}(\Sigma_g)$...