  Metacirculants are a basic and well-studied family of vertex-transitive
graphs, and weak metacirculants are generalizations of them. A graph is called
a weak metacirculant if it has a vertex-transitive metacyclic automorphism
group. This paper is devoted to the study of weak metacirculants with odd prime
power order. We first prove that a weak metacirculant of odd prime power order
is a metacirculant if and only if it has a vertex-transitive split metacyclic
automorphism group. We then prove that for any odd prime $p$ and integer
$\ell\geq 4$, there exist weak metacirculants of order $p^\ell$ which are
Cayley graphs but not Cayley graphs of any metacyclic group; this answers a
question in Li et al. (2013). We construct such graphs explicitly by
introducing a construction which is a generalization of generalized Petersen
graphs. Finally, we determine all smallest possible metacirculants of odd prime
power order which are Cayley graphs but not Cayley graphs of any metacyclic
group.
