In US copyright law, this is called a “mechanical license”. The cover artist pays a fee to a licensing agency, which compensates the original artist. They don’t have to get the original artist’s consent; rather, paying the license fee entitles the cover artist to record and sell the cover version.
(It’s called “mechanical” because it was originally devised for player pianos.)
https://en.wikipedia.org/wiki/Mechanical_license
https://en.wikipedia.org/wiki/Compulsory_license
https://en.wikipedia.org/wiki/Cover_version#United_States_copyright_law
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