  We have used optical Integral Field Spectroscopy in order to map the star
formation history of the inner kiloparsec of the Post-Starburst Quasar (PSQ)
J0330--0532 and to map its gas and stellar kinematics as well as the gas
excitation. PSQs are hypothesized to represent a stage in the evolution of
galaxies in which the star formation has been recently quenched due to the
feedback of the nuclear activity, as suggested by the presence of the
post-starburst population at the nucleus. We have found that the old stellar
population (age $\ge$ 2.5 Gyr) dominates the flux at 5100 \AA\ in the inner
0.26 kpc, while both the post-starburst (100 Myr $\le$ age $<$ 2.5 Gyr) and
starburst (age $<$ 100 Myr) components dominate the flux in a circumnuclear
ring at $\approx$0.5 kpc from the nucleus. With our spatially resolved study we
do not have found any post-starburst stellar population in the inner 0.26\,kpc.
On the other hand, we do see the signature of AGN feedback in this region,
which does not reach the circumnuclear ring where the post-starburst population
is observed. We thus do not support the quenching scenario for the J0330-0532.
In addition, we have concluded that the strong signature of the post-starburst
population in larger aperture spectra (e.g. from Sloan Digital Sky Survey) is
partially due to the combination of the young and old age components. Based on
the M$_{\rm BH}-\sigma_{\rm star}$ relationship and the stellar kinematics we
have estimated a mass for the supermassive black hole of 1.48 $\pm$ 0.66
$\times$10$^7$ M$_\odot$.
