  For a finite group, it is interesting to determine when two ordinary
irreducible representations have the same $p$-modular reduction; that is, when
two rows of the decomposition matrix in characteristic $p$ are equal, or
equivalently when the corresponding $p$-modular Brauer characters are the same.
We complete this task for the double covers of the symmetric group when $p=2$,
by determining when the $2$-modular reduction of an irreducible spin
representation coincides with a $2$-modular Specht module. In fact, we obtain a
more general result: we determine when an irreducible spin representation has
$2$-modular Brauer character proportional to that of a Specht module. In the
course of the proof, we use induction and restriction functors to construct a
function on generalised characters which has the effect of swapping runners in
abacus displays for the labelling partitions.
