Why did you catch a cold and your hands and feet itch?
First of all, if the hands and feet are frozen, the blood vessels here will contract, and the blood flow through here will slow down and the blood flow will decrease. In this way, the energy available to hands and feet will be much reduced. Through the adjustment of nerves and body fluids, the body will feel itchy, which will make you have the urge to scratch. If you scratch or rub it, it will heat the area and the blood vessels will expand accordingly, thus increasing the blood flow here.
Of course, if you freeze too much, blood vessels will also be damaged. Such scratching will dilate the blood vessels in the frostbite area, and inflammatory exudation will swell the affected area. This is chilblain.
But some people don't feel itchy when they are badly frozen. This is because of their own physical problems.