On endomorphisms of the Cuntz algebra which preserve the canonical UHF-subalgebra, II

It was shown recently by Conti, Rørdam and Szymański that there exist endomorphisms λu of the Cuntz algebra On such that λu(Fn)⊆Fn but u∉Fn, and a question was raised if for such a u there must always exist a unitary v∈Fn with λu|Fn=λv|Fn. In the present paper, we answer this question to the negative. To this end, we analyze the structure of such endomorphisms λu for which the relative commutant λu(Fn)′∩Fn is finite dimensional.