Had a Datsun 280 zx that did just that. Replaced outer tie rods, as those were loose, as well as sway bar bushings. Neither fixed problem. New tires and balance took care of 45-55 shake. For now rotate rear to front tires, see what happens. That won't cost a dime. If it takes care of it replace tires in 5 k when it's time to rotate. Assuming tires are same size front to rear.
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