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

Statement
Suppose $$G$$ is a fact about::finite group that is a fact about::rational group and $$P$$ is a $$2$$-Sylow subgroup of $$G$$. Further, suppose $$P$$ is a fact about::group of nilpotency class two, i.e., the nilpotency class of $$P$$ is at most two. Then, $$P$$ is also a proves property satisfaction of::rational group.

Related facts

 * 2-Sylow subgroup of rational group need not be rational