  Although Sarnak's conjecture holds for compact group rotations (irrational
rotations, odometers), it is not even known whether it holds for all
Jewett-Krieger models of such rotations. In this paper we show that it does, as
long as the model is at the same a topological extension. In particular, we
reestablish (after [AKL]) that regular Toeplitz systems satisfy Sarnak's
conjecture, and, as another consequence, so do all generalized Sturmian
subshifts (not only the classical Sturmian subshift). We also give an example
of an irregular Toeplitz subshift which fits our criterion. We give an example
of a model of an odometer which is not even Toeplitz (it is weakly mixing),
hence does not fit our criterion. However, for this example, we manage to
produce a separate proof of Sarnak's conjecture. Next, we provide a class of
Toeplitz sequences which fail Sarnak's conjecture (in a weak sense); all these
examples have positive entropy. Finally, we examine the example of a Toeplitz
sequence from [AKL] (which fails Sarnak's conjecture in the strong sense) and
prove that it has positive entropy, as well (this proof has been announced in
[AKL]). This paper can be considered a sequel to [AKL], it also fills some gaps
of [D].
