  We present Chandra imaging and spectral observations of Seyfert 2 galaxy NGC
4388. Three extended X-ray structures around the nucleus on kpc scale are well
imaged, allowing an in-depth spatially resolved study. Both the extended hard
continuum and the Fe K$\alpha$ line show similar morphology, consistent with a
scenario where the ionizing emission from nucleus is reprocessed by
circumnuclear cold gas, resulting in a weak reflection continuum and an
associated neutral Fe K$\alpha$ line. This has been seen in other Compton thick
active galactic nuclei (AGN), but NGC 4388 is one of the rare cases with a
lower column density ($N_{\rm H} < 1.25\times 10^{24}$ cm$^{-2}$) along the
line of sight. Significant differences in equivalent width of the Fe K$\alpha$
emission line are found for the nuclear and extended regions, which could be
ascribed to different column densities or scattering angles with respect to the
line of sight, rather than variations in iron abundances. The north-east and
west extended structures are aligned with the galactic disk and dust lane in
the HST $V-H$ map, and located at the peak of molecular gas distribution. The
morphology implies that the kpc-scale radio outflow may have compressed the
interstellar gas and produced clumps working as the reflector to enhance line
emission. Using [OIV] emission as a proxy of the AGN intrinsic luminosity, we
find that both of the extended Fe K$\alpha$ emission and reflection continuum
are linearly correlated with the [OIV] luminosity, indicating a connection
between the AGN and the extended emission.
