  For any finite group $G$ and any prime $p$ one can ask which ordinary
irreducible representations remain irreducible in characteristic $p$, or more
generally, which representations remain homogeneous in characteristic $p$. In
this paper we address this question for $p=3$ when $G$ is a proper double cover
of the symmetric or alternating group. We obtain a classification except in the
case of a certain family of partitions relating to spin RoCK blocks. Our
techniques involve induction and restriction, degree calculations, decomposing
projective characters and recent results of Kleshchev and Livesey on spin RoCK
blocks.
