The flag was actually tied to a dead tree limb. The subsequent peace treaty which ended the war transferred Wake to the United States. Wake does not have and never has had a native population. The closest thing to it are the Chamorro employees of Pan American Airways, natives of Guam and American citizens who resided there from 1935 to 1942. It has only been continuously occupied since 1935, by Americans with a short interval of Japanese occupation.
Wake Island is indisputably American, paid for in blood.
Any other claim is specious, ridiculous, and utterly false.