  We study the extensions of teleparallism in the Kaluza-Klein (KK) scenario by
writing the analogous form to the torsion scalar $T_{\text{NGR}}$ in terms of
the corresponding antisymmetric tensors, given by $T_{\text{NGR}} = a\,T_{ijk}
\, T^{ijk} + b\,T_{ijk} \,T^{kji} + c\,T^{j}{}_{ji} \, T^{k}{}_{k}{}^{i}$, in
the four-dimensional New General Relativity (NGR) with arbitrary coefficients
$a$, $b$ and $c$. After the KK dimensional reduction, the Lagrangian in the
Einstein-frame can be realized by taking $2a+b+c=0$ with the ghost-free
condition $c\leq0$ for the one-parameter family of teleparallelism. We
demonstrate that the pure conformal invariant gravity models can be constructed
by the requirements of $2a+b=0$ and $c=0$. In particular, the torsion vector
can be identified as the conformal gauge field, while the conformal gauge
theory can be obtained by $2a+b+4c=0$ or $2a+b=0$, which is described on the
Weyl-Cartan geometry $Y_4$ with the ghost-free conditions $2a+b+c>0$ and
$c\neq0$. We also consider the weak field approximation and discuss the
non-minimal coupled term of the scalar current and torsion vector. For the
conformal invariant models with $2a+b=0$, we find that only the anti-symmetric
tensor field is allowed rather than the symmetric one.
