  We have constructed a spherically symmetric structure model in a cosmological
background filled with perfect fluid with non-vanishing pressure as an exact
solution of Einstein equations using the Lema\^{i}tre solution. To study its
local and quasi-local characteristics including the novel features of its
central black hole, we have suggested an algorithm to integrate the equations
numerically. The result shows intriguing effects of the pressure inside the
structure. The evolution of the central black hole within the FRW universe, its
decoupling from the expanding parts of the model, the structure of its
space-like apparent horizon, the limiting case of the dynamical horizon tending
to a slowly evolving horizon, and the decreasing mass in-fall to the black hole
is also studied. The cosmological redshift of a light emitted from the
cosmological structure to an observer in the FRW background is also calculated.
This cosmological redshift includes local and cosmic part which are explicitly
separated. We have also formulated a modified NFW density profile for a
structure to match the exact solution conditions.
