  We relate the linear asymptotic representation theory of the symmetric groups
to its spin counterpart. In particular, we give explicit formulas which express
the normalized irreducible spin characters evaluated on a strict partition
$\xi$ with analogous normalized linear characters evaluated on the double
partition $D(\xi)$. We also relate some natural filtration on the usual
(linear) Kerov-Olshanski algebra of polynomial functions on the set of Young
diagrams with its spin counterpart. Finally, we give a spin counterpart to
Stanley formula for the characters of the symmetric groups.
