For a metric space $(X,d)$, let $\def\Iso{\operatorname{Iso}}\Iso(X,d)$ denote the group of bijective isometries of $(X,d)$. Clearly, $\Iso(X,d)$ is a group under composition.
Question: Let $X$ be a space with two equivalent metrics $d_1$ and $d_2$. Is it true that $\Iso(X,d_1)$ and $\Iso(X,d_2)$ are isomorphic?
Consider the set $X=\{0,1,2\}$ with the metric $d$ inherited from the usual Euclidean metric and the metric $d'$ defined by
$$d'(x,y)=\begin{cases} 0,&\text{if }x=y\\ 1,&\text{if }x\ne y\;. \end{cases}\tag{1}$$
Every permutation of $X$ is in $\operatorname{Iso}(X,d')$, but $\operatorname{Iso}(X,d)$ is isomorphic to $\Bbb Z/2\Bbb Z$.
For an infinite example, let $X=\{2^n:n\in\Bbb N\}$, let $d$ be the usual metric, and let $d'$ be the discrete metric as in $(1)$: $\operatorname{Iso}(X,d)$ is trivial, while $\operatorname{Iso}(X,d')$ is isomorphic to the group of permutations of $\Bbb N$.
