Endomorphisms of On which preserve the canonical UHF-subalgebra

Unital endomorphisms of the Cuntz algebra On which preserve the canonical UHF-subalgebra Fn⊆On are investigated. We give examples of such endomorphisms λ=λu for which the associated unitary element u in On (which satisfies λ(Sj)=uSj for all j) does not belong to Fn. One such example, in the case where n=2, arises from a construction of a unital endomorphism on O2 which preserves the canonical UHF-subalgebra and where the relative commutant of its image in O2 contains a copy of O2.