  Let $m,n,r$ be positive integers, and let $G=\langle a\rangle: \langle
b\rangle \cong \mathbb{Z}_n: \mathbb{Z}_m$ be a split metacyclic group such
that $b^{-1}ab=a^r$. We say that $G$ is {\em absolutely split with respect to
$\langle a\rangle$} provided that for any $x\in G$, if $\langle
x\rangle\cap\langle a\rangle=1$, then there exists $y\in G$ such that
$x\in\langle y\rangle$ and $G=\langle a\rangle: \langle y\rangle$. In this
paper, we give a sufficient and necessary condition for the group $G$ being
absolutely split. This generalizes a result of Sanming Zhou and the second
author in [arXiv: 1611.06264v1]. We also use this result to investigate the
relationship between metacirculants and weak metacirculants.
  Metacirculants were introduced by Alspach and Parsons in $1982$ and have been
a rich source of various topics since then. As a generalization of this classes
of graphs, Maru\v si\v c and \v Sparl in 2008 posed the so called weak
metacirculants. A graph is called a {\em weak metacirculant} if it has a
vertex-transitive metacyclic automorphism group. In this paper, it is proved
that a weak metacirculant of $2$-power order is a metacirculant if and only if
it has a vertex-transitive split metacyclic automorphism group. This provides a
partial answer to an open question in the literature.
