  Joyce, Tanaka, and Upmeier give an orientation of the $G$-instanton moduli
spaces on a closed four manifolds which is canonically defined using the the
$\mathrm{Spin}^c$ structure on the $4$-manifold. In this note, we describe the
relation between the orientations given by other choices of the
$\mathrm{Spin}^c$ structures, in a slightly more general setting. Furthermore,
we give an alternative proof of the orientability of the instanton moduli
spaces and an alternative construction of the orientation by Joyce, Tanaka and
Upmeier, by using Witten localization of the index of a Dirac type operator.
