Promote the majority to the extreme (11)
I believe the answer is:
furthermost
'the extreme' is the definition.
(thesaurus)
'promote the majority' is the wordplay.
'promote' becomes 'further' (furthering is a kind of promoting).
'the majority' becomes 'most' (most of a group).
'further'+'most'='FURTHERMOST'
'to' acts as a link.
(Other definitions for furthermost that I've seen before include "extremely remote" , "The greatest distance away" .)