We investigate the relationship between endomorphisms of the Cuntz algebra O2 and endomorphisms of the Thompson groups F , T and V represented inside the unitary group of O2. For an endomorphism λu of O2, we show that λu(V ) ⊆ V if and only if u ∈ V . If λu is an automorphism of O2 then u ∈ V is equivalent to λu(F ) ⊆ V . Our investigations are facilitated by introduction of the concept of modestly scaling endomorphism of On, whose properties and examples are investigated.