Let $(X,d)$ be a metric space, whose metric $d$ is not known.
Let $G=(f,\circ)$ be its group of isometries (that is, distance preserving functions $f:X\rightarrow X$, with the usual function composition as group operation).
Is $d$ uniquely specified by $G$?
If the answer is yes: how can we explicitely know the form of $d$ from $G$?
If the answer is no: under what simplifying assumptions will $d$ be specified by $G$?
No. If $d$ is a metric, then so is $d/(1+d)$. There are probably many functions other than $x\mapsto x/(1+x)$ that could be composed with a metric to produce another metric.
So the best you might hope for, is that two metrics with the same isometries are related by composition with some function. Whether that is true or not, I don't know.