  Carlsson's construction is a simplicial group whose geometric realization is
the loop space of the 1-stunted reduced Borel construction. Our main results
are: i) Given a pointed simplicial set acted upon by the discrete cyclic group
C_2 of order 2, if the orbit projection has a section, then this loop space has
a mod 2 homology decomposition; ii) If the reduced diagonal map of the
C_2-invariant set is homologous to zero, then the pinched sets in the above
homology decomposition themselves have homology decompositions in terms of the
C_2-invariant set and the orbit space. Result i) generalizes a previous
homology decomposition of the second author for trivial actions. To illustrate
these two results, we completely compute the mod 2 Betti numbers for an
example.
