  We study the asymptotics of iterates of the transfer operator for
non-uniformly hyperbolic $\alpha$-Farey maps. We provide a family of
observables which are Riemann integrable, locally constant and of bounded
variation, and for which the iterates of the transfer operator, when applied to
one of these observables, is not asymptotic to a constant times the wandering
rate on the first element of the partition $\alpha$. Subsequently, sufficient
conditions on observables are given under which this expected asymptotic holds.
In particular, we obtain an extension theorem which establishes that, if the
asymptotic behaviour of iterates of the transfer operator is known on the first
element of the partition $\alpha$, then the same asymptotic holds on any
compact set bounded away from the indifferent fixed point.
