  The representation theory of 0-Hecke-Clifford algebras as a degenerate case
is not semisimple and also with rich combinatorial meaning. Bergeron et al.
have proved that the Grothendieck ring of the category of finitely generated
supermodules of 0-Hecke-Clifford algebras is isomorphic to the algebra of peak
quasisymmetric functions defined by Stembridge. In this paper we further study
the category of finitely generated projective supermodules and clarify the
correspondence between it and the peak algebra of symmetric groups. In
particular, two kinds of restriction rules for induced projective supermodules
are obtained. After that, we consider the corresponding Heisenberg double and
its Fock representation to prove that the ring of peak quasisymmetric functions
is free over the subring of symmetric functions spanned by Schur's Q-functions.
