Group that counts Harding, Roosevelt, Truman and Kennedy as former members (4)

I believe the answer is:

elks

(Other definitions for elks that I've seen before include "Moose(s)" , "Large deer of Europe and N America" , "Large deer of Northern Europe" , "Moose" , "Large deer with flattened antlers" .)

I've seen this clue in The New York Times.

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