  For any finite group $G$ and any prime $p$ one can ask which ordinary
irreducible representations remain irreducible in characteristic $p$. We answer
this question for $p=2$ when $G$ is a proper double cover of the symmetric
group. Our techniques involve constructing part of the decomposition matrix for
a Rouquier block of a double cover, restricting to subgroups using the
Brundan--Kleshchev modular branching rules and comparing the dimensions of
irreducible representations via the bar-length formula.
