  In general relativity coupled to Maxwell's electromagnetism and charged
matter, when the gravitational potential $W^2$ and the electric potential field
$\phi$ obey a relation of the form $W^{2}= a\left(-\epsilon\, \phi+ b\right)^2
+c$, where $a$, $b$ and $c$ are arbitrary constants, and $\epsilon=\pm1$ (the
speed of light $c$ and Newton's constant $G$ are put to one), a class of very
interesting electrically charged systems with pressure arises. We call the
relation above between $W$ and $\phi$, the Weyl-Guilfoyle relation, and it
generalizes the usual Weyl relation, for which $a=1$. For both, Weyl and
Weyl-Guilfoyle relations, the electrically charged fluid, if present, may have
nonzero pressure. Fluids obeying the Weyl-Guilfoyle relation are called
Weyl-Guilfoyle fluids. These fluids, under the assumption of spherical
symmetry, exhibit solutions which can be matched to the electrovacuum
Reissner-Nordstr\"om spacetime to yield global asymptotically flat cold charged
stars. We show that a particular spherically symmetric class of stars found by
Guilfoyle has a well-behaved limit which corresponds to an extremal
Reissner-Nordstr\"om quasiblack hole with pressure, i.e., in which the fluid
inside the quasihorizon has electric charge and pressure, and the geometry
outside the quasihorizon is given by the extremal Reissner-Nordstr\"om metric.
The main physical properties of such charged stars and quasiblack holes with
pressure are analyzed. An important development provided by these stars and
quasiblack holes is that without pressure the solutions, Majumdar-Papapetrou
solutions, are unstable to kinetic perturbations. Solutions with pressure may
avoid this instability. If stable, these cold quasiblack holes with pressure,
i.e., these compact relativistic charged spheres, are really frozen stars.
