  We extend the study of the core of the Fe K$\alpha$ emission line at \sim 6.4
keV in Seyfert galaxies reported in Yaqoob & Padmanabhan (2004) using a larger
sample observed by the Chandra High Energy Grating (HEG). Whilst heavily
obscured active galactic nuclei (AGNs) are excluded from the sample, these data
offer some of the highest precision measurements of the peak energy of the Fe
K$\alpha$ line, and the highest spectral resolution measurements of the width
of the core of the line in unobscured and moderately obscured ($N_{H}<10^{23} \
\rm cm^{-2}$) Seyfert galaxies to date. The Fe K$\alpha$ line is detected in 33
sources, and its centroid energy is constrained in 32 sources. In 27 sources
the statistical quality of the data is good enough to yield measurements of the
FWHM. We find that the distribution in the line centroid energy is strongly
peaked around the value for neutral Fe, with over 80% of the observations
giving values in the range 6.38--6.43 keV. Including statistical errors, 30 out
of 32 sources ($\sim 94%$) have a line centroid energy in the range 6.35--6.47
keV. The mean equivalent width, amongst the observations in which a non-zero
lower limit could be measured, was $53 \pm 3eV. The mean FWHM from the
subsample of 27 sources was $2060 \pm 230 \ \rm km \ s^{-1}$. The mean EW and
FWHM are somewhat higher when multiple observations for a given source are
averaged. From a comparison with the H$\beta$ optical emission-line widths (or,
for one source, Br$\alpha$), we find that there is no universal location of the
Fe K$\alpha$ line-emitting region relative to the optical BLR. We confirm the
presence of the X-ray Baldwin effect, an anti-correlation between the Fe
K$\alpha$ line EW and X-ray continuum luminosity. The HEG data have enabled
isolation of this effect to the narrow core of the Fe K$\alpha$ line.
