  We investigate the relationship between endomorphisms of the Cuntz algebra
${\mathcal O}_2$ and endomorphisms of the Thompson groups $F$, $T$ and $V$
represented inside the unitary group of ${\mathcal O}_2$. For an endomorphism
$\lambda_u$ of ${\mathcal O}_2$, we show that $\lambda_u(V)\subseteq V$ if and
only if $u\in V$. If $\lambda_u$ is an automorphism of ${\mathcal O}_2$ then
$u\in V$ is equivalent to $\lambda_u(F)\subseteq V$. Our investigations are
facilitated by introduction of the concept of modestly scaling endomorphism of
${\mathcal O}_n$, whose properties and examples are investigated.
