The group acts on an ordered set

Question

Let $G$ be a group, $G$ acts on an ordered set and preserves its order, i.e. $a<b$, then $g(a)<g(b)$ for $g\in G$. Then does it imply there is a left order on $G$, i.e. $f<g$, then $fh<gh$ for $h\in G$?

Answer 1

No. Form a diamond with $M$ the unique maximum, $m$ the unique minimum, and $x,y$ two incomparable elements that are both in between $M$ and $m$. We can let $G=C_2$ act by switching $x$ and $y$ and leaving the max/min fixed. This preserves the ordering, but there is no possible ordering on $G$ compatible with the group operation.