  In this paper, we investigate the properties of $\sigma$-A-nuclei of a
quasigroup including relations between them and relations between their
respective component sets, where $\sigma \in S_3$. We also find connections
between components of $\sigma$-A-nuclei of a quasigroup and components of
$\sigma$-A-nuclei of the isostrophic images. Further, we investigate the
properties of various inverse quasigroups using the derived connections. These
properties will not only make the study of $\sigma$-A-nuclei of a quasigroup
simple but also reduce the time required in the computation of
$\sigma$-A-nuclei of a quasigroup for different values of $\sigma$.
