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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. We have also calculated the redshift of a light emitted from nearby the cosmological structure to an observer in the FRW background and have shown that it contains both the local gravitational and the cosmological redshift with some observational consequences. It has been shown that this type of cosmological black holes have the flexibility to match with the NFW dark matter density profile.
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