Let $X$ be a topological space and let $\mathcal{F}$ be a collection of homeomorphisms of $X$. I would like to investigate the possibility of defining a metric $d$ on $X$ such that the the isometries of the metric space $(X,d)$ is precisely the set $\mathcal{F}$. Note that I want the topology induced by $d$ to coincide with the original topology on $X$. What are the conditions on $X$ and $\mathcal{F}$ that allow us to construct such a $d$?

I am interested in this question from the perspective of hyperbolic geometry. On the unit disk the Poincare metric (up to a constant) is the unique metric that has isometry group the holomorphic automorphisms and conjugates. I want to find a metric on $\mathbb{C} \setminus \{0,1\}$ such that the isometries are precisely the holomorphic automorphisms of $\mathbb{C} \setminus \{0,1\}$ and the conjugates of these automorphisms. The Kobayashi pseudodistance on $\mathbb{C} \setminus \{0,1\}$ is certainly one distance that satisfies this condition but the fact that $\mathbb{C} \setminus \{0,1\}$ is Kobayashi hyperbolic is a non-trivial fact. I want to see whether it possible show the existence of such a metric by other means. I also want to see how this new metric is related to the Kobayashi distance

Let $X$ be a (sufficiently nice) topological space and let $\mathcal{F}$ be a group of homeomorphisms of $X$. Assume that $\mathcal{F}$ is also closed under point-wise convergence. I would like to investigate the possibility of defining a metric $d$ on $X$ such that the the isometries of the metric space $(X,d)$ is precisely the set $\mathcal{F}$. Note that I want the topology induced by $d$ to coincide with the original topology on $X$. What are the conditions on $X$ and $\mathcal{F}$ that allow us to construct such a $d$?

I am interested in this question from the perspective of hyperbolic geometry. On the unit disk the Poincare metric (up to a constant) is the unique metric that has isometry group the holomorphic automorphisms and conjugates. I want to find a metric on $\mathbb{C} \setminus \{0,1\}$ such that the isometries are precisely the holomorphic automorphisms of $\mathbb{C} \setminus \{0,1\}$ and the conjugates of these automorphisms. The Kobayashi pseudodistance on $\mathbb{C} \setminus \{0,1\}$ is certainly one distance that satisfies this condition but the fact that $\mathbb{C} \setminus \{0,1\}$ is Kobayashi hyperbolic is a non-trivial fact. I want to see whether it possible show the existence of such a metric by other means. I also want to see how this new metric is related to the Kobayashi distance