2-Sylow subgroup of rational group need not be rational

Statement
It is possible to have a finite group $$G$$ such that $$G$$ is a fact about::rational group and $$G$$ has a $$2$$-fact about::Sylow subgroup $$P$$ that is not a fact about::rational group.

Related facts

 * 2-Sylow subgroup of rational group is rational if its class is at most two