Any free presentation for the finite group
G
G
determines a central extension
(
R
,
F
)
(R,F)
for
G
G
having the projective lifting property for
G
G
over any field
k
k
. The irreducible representations of
F
F
which arise as lifts of irreducible projective representations of
G
G
are investigated by considering the structure of the group algebra
k
F
kF
, which is greatly influenced by the fact that the set of torsion elements of
F
F
is equal to its commutator subgroup and, in particular, is finite. A correspondence between projective equivalence classes of absolutely irreducible projective representations of
G
G
and
F
F
-orbits of absolutely irreducible characters of
F
′
F’
is established and employed in a discussion of realizability of projective representations over small fields.