Primes modulo which almost all Fermat numbers are primitive roots

A prime p is called elite, or anti-elite, when all but finitely many Fermat numbers are quadratic nonresidues or residues, respectively, modulo p. It is known that if the multiplicative order of 2 modulo p is of the form 2(s) x 5, where s >= 2, then the prime p is either elite or anti-elite. Modulo elite primes of this kind, we describe some criteria by which all sufficiently large Fermat numbers be primitive roots, or all nonprimitive roots.