Consider the group of integers $\mathbb{Z}$ equipped with the discrete metric $$d(m, n) = \begin{cases} 1, \quad m \neq n\\ 0 , \quad m = n \end{cases}$$ In particular, $\mathbb{Z}$ is a metric Lie group with left-invariant metric $d$.
I would like to study the action of $\text{Iso}(\mathbb{Z})$ on $\mathbb{Z}$ $$\text{Iso}(\mathbb{Z}) \, \times \, \mathbb{Z} \rightarrow \mathbb{Z}, \quad (F, \, m) \, \mapsto \, F(m),$$ where $\text{Iso}(\mathbb{Z})$ is the isometry group of $\mathbb{Z}$.
I'm not able to visualize in a concrete way what gives $\text{Iso}(\mathbb{Z})$ in this context in order to try after to prove a result about the stabilizers of $\text{Iso}(\mathbb{Z})$.
Any suggestions? Thanks in advance!