  We thoroughly investigate conformally Schwarzschild spacetimes in different
coordinate systems to seek for physically reasonable models of a cosmological
black hole. We assume that a conformal factor depends only on the time
coordinate and that the spacetime is asymptotically flat
Friedmann-Lema\^{\i}tre-Robertson-Walker universe filled by a perfect fluid
obeying a linear equation state $p=w\rho$ with $w>-1/3$. In this class of
spacetimes, the McClure-Dyer spacetime, constructed in terms of the isotropic
coordinates, and the Thakurta spacetime, constructed in terms of the standard
Schwarzschild coordinates, are identical and do not describe a cosmological
black hole. In contrast, the Sultana-Dyer and Culetu classes of spacetimes,
constructed in terms of the Kerr-Schild and Painlev\'{e}-Gullstrand
coordinates, respectively, describe a cosmological black hole. In the
Sultana-Dyer case, the corresponding matter field in general relativity can be
interpreted as a combination of a homogeneous perfect fluid and an
inhomogeneous null fluid, which is valid everywhere in the spacetime unlike
Sultana and Dyer's interpretation. In the Culetu case, the matter field can be
interpreted as a combination of a homogeneous perfect fluid and an
inhomogeneous anisotropic fluid. However, in both cases, the total
energy-momentum tensor violates all the standard energy conditions at a finite
value of the radial coordinate in late times. As a consequence, the
Sultana-Dyer and Culetu black holes for $-1/3<w\le 1$ cannot describe the
evolution of a primordial black hole after its horizon entry.
