Kerov polynomials describe normalized irreducible characters of the symmetric
groups in terms of the free cumulants associated with Young diagrams. We
suggest well-suited counterparts of the Kerov polynomials in spin (or
projective) representation settings. We show that spin analogues of irreducible
characters are polynomials in even free cumulants associated with double
diagrams of strict partitions. Moreover, we present a conjecture for the
positivity of their coefficients
