If a manifold $M^n$ is given then we know how to find a symmetry group(Automorphism group) of $M^n$ which will preserve some objects(like metric, connections etc) related to $M^n$.
But my query here is the converse of this that if some lie group $G$ is given, then how can we construct the possible manfolds $M^n$ whose automorphism group will be the given group $G$.
Suppose we are given a $n$-dimensional lie group $G$ which is inducing a set of diffeomorphisms on some open sets of $\mathbb{R}^n$. $$x \longrightarrow x' = \underbrace{g\big(\{\theta^\nu \}_{\nu=1}^{n}\big)}_{\in \hspace{1mm} G} \quad x$$ where $\theta^\nu $ are the group parameters of $G$; $\quad x= (x^1,x^2,...,x^n)$, $x'= ({x'}^1,{x'}^2,...,{x'}^n)$
and $x,x' \in \mathbb{R}^n$.
Now there will always exist atleast a metric spaced manifold($M^n$) such that the automorphism group of $M^n$ is the given group $G$, subject to some invariance conditions like the diffeomorphisms of $M$ induced by $G$ will preserve the metric, affine connection etc.
For example,
Galilean group $G(d,1)$ acting as a diffeomorphism on $\mathbb{R}^{d+1}$ as,
\begin{align} x^i & \longrightarrow {x'}^i = {R}^{i}_{\hspace{1mm} j}x^j + b^i x^1 + > c^i; \quad \quad \forall i=1,2,3,...,d \\ x^0 & \longrightarrow x^0 + e \end{align} where $R \in $ SO($d$), $\vec{b}$ & $\vec{c} \in \mathbb{R}^d$, $e \in \mathbb{R}$
Then, a smooth Non-Riemannian manifold($N^{d+1},\gamma,\theta,\nabla$) where $\gamma$ is a contravariant, twice-symmetric,positive tensor field and degenerate metric; $\theta$ is a 1-form which generates the kernel of $\gamma$; $\nabla$ is a symmetric affine connection that parallel transports both $\gamma$ & $\theta$.
$N^{d+1}$ is constructed as: \begin{align} N^{d+1} & = \mathbb{R} \times \mathbb{R}^{d} \\ \gamma & = \delta^{ij} \frac{\partial}{\partial x^i} \otimes \frac{\partial}{\partial x^j}; \quad \quad \forall i,j = 1,2,3,...,d \\ \theta & = dt \\ \Gamma^{\rho}_{\hspace{1mm}\mu\nu} & = 0; \quad \quad \quad \quad \quad \quad \quad \forall \mu,\nu,\rho = 0,1,2,3,...,d \end{align} Now, the automorphisms of ($N^{d+1},\gamma,\theta,\nabla$) i.e. the transformations which preserve all the geometric ingredients $\gamma,\theta,\nabla$ constitute the Galilean Group$(G(d,1))$.
Now, from knowing the $G(d,1)$ and its diffeomorphic action on open sets of $\mathbb{R}^{d+1}$, how to arrive at the manifold structure($N^{d+1},\gamma,\theta,\nabla$)?
Many such constructions of manifolds from a given symmetry group can be found in Duval, C.; Gibbons, G. W.; Horvathy, P. A.; Zhang, P. M., Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time, Classical Quantum Gravity 31, No. 8, Article ID 085016, 24 p. (2014). ZBL1295.83006.
Reframing my question,
Question: What are the methods to construct such manifolds $M^n$ mentioned above, if I know the action of $G$ as diffeomorphisms on some open sets of $\mathbb{R}^n$?
(P.S. I will be glad to hear from the respected community and also it will be helpful if you kindly mention how the construcion of the manifolds $M^n$ will change if the 'subject to some invariance conditions' mentioned above are also altered.)