  It was shown recently by Conti, R{\o}rdam and Szyma\'{n}ski that there exist
endomorphisms $\lambda_u$ of the Cuntz algebra $\mathcal{O}_n$ such that
$\lambda_u (\mathcal{F}_n)\subseteq\mathcal{F}_n$ but $u\not\in\mathcal{F}_n$,
and a question was raised if for such a $u$ there must always exist a unitary
$v\in\mathcal{F}_n$ with $\lambda_u|_{\mathcal{F}_n} =
\lambda_v|_{\mathcal{F}_n}$. In the present paper, we answer this question to
the negative. To this end, we analyze the structure of such endomorphisms
$\lambda_u$ for which the relative commutant
$\lambda_u(\mathcal{F}_n)'\cap\mathcal{F}_n$ is finite dimensional.
