  Discrete spacetime symmetries of parity P or reflection R, and time-reversal
T, act naively as $\mathbb{Z}_2$-involutions in the passive transformation on
the spacetime coordinates; but together with a charge conjugation C, the total
C-P-R-T symmetries have enriched active transformations on fields in
representations of the spacetime-internal symmetry groups of quantum field
theories (QFTs). In this work, we derive that these symmetries can be further
fractionalized, especially in the presence of the fermion parity $(-1)^{\rm
F}$. We elaborate on examples including relativistic Lorentz invariant QFTs
(e.g., spin-1/2 Dirac or Majorana spinor fermion theories) and nonrelativistic
quantum many-body systems (involving Majorana zero modes), and comment on
applications to spin-1 Maxwell electromagnetism (QED) or interacting Yang-Mills
(QCD) gauge theories. We discover various C-P-R-T-$(-1)^{\rm F}$ group
structures, e.g., Dirac spinor is in a projective representation of
$\mathbb{Z}_2^{\rm C}\times \mathbb{Z}_2^{\rm P} \times \mathbb{Z}_2^{\rm T}$
but in an (anti)linear representation of an order-16 nonabelian finite group,
as the central product between an order-8 dihedral (generated by C and P) or
quaternion group and an order-4 group generated by T with T$^2=(-1)^{\rm F}$.
The general theme may be coined as C-P-T or C-R-T fractionalization.
