A twisted cocyle with values in a Lie group $G$ is a cocyle
that incorporates an automorphism of $G$.
Suppose that the underlying transformation is hyperbolic.
We prove that if two Hölder continuous twisted cocycles with
a sufficiently high Hölder exponent assign equal 'weights'
to the periodic orbits of $\phi$, then they are Hölder
cohomologous. This generalises a well-known theorem
due to Livšic in the untwisted case. Having determined
conditions for there to be a solution to the twisted
cocycle equation, we consider how many other solution
there may be. When $G$ is a toius, we determine conditions
for there to be only finitely many solutions to the
twisted cocycle equation.