  Twisted homomorphisms of bialgebras are bialgebra homomorphisms from the
first into Drinfeld twistings of the second. They possess a composition
operation extending composition of bialgebra homomorphisms. Gauge
transformations of twists, compatible with adjacent homomorphisms, give rise to
gauge transformation of twisted homomorphisms, which behave nicely with respect
to compositions. Here we study (gauge classes of) twisted automorphisms of
cocommutative Hopf algebras. After revising well-known relations between
twists, twisted forms of bialgebras and $R$-matrices (for commutative
bialgebras) we describe twisted automorphisms of universal enveloping algebras.
