  This paper considers the design of nonlinear observers for invariant systems
posed on finite-dimensional connected Lie groups with measurements generated by
a transitive group action on an associated homogeneous space. We consider the
case where the group action has the opposite invariance to the system
invariance and show that the group kinematics project to a minimal realisation
of the systems observable dynamics on the homogeneous output space. The
observer design problem is approached by designing an observer for the
projected output dynamics and then lifting to the Lie-group. A structural
decomposition theorem for observers of the projected system is provided along
with characterisation of the invariance properties of the associated observer
error dynamics. We propose an observer design based on a gradient-like
construction that leads to strong (almost) global convergence properties of
canonical error dynamics on the homogeneous output space. The observer dynamics
are lifted to the group in a natural manner and the resulting gradient-like
error dynamics of the observer on the Lie-group converge almost globally to the
unobservable subgroup of the system, the stabiliser of the group action.
