  In this short paper we look at the action of T-duality and string duality
groups on fermions, in maximally-supersymmetric theories and related theories.
Briefly, we argue that typical duality groups such as SL(2,Z) have sign
ambiguities in their actions on fermions, and propose that pertinent duality
groups be extended by Z_2, to groups such as the metaplectic group.
Specifically, we look at duality groups arising from mapping class groups of
tori in M theory compactifications, T-duality, ten-dimensional type IIB
S-duality, and (briefly) four-dimensional N=4 super Yang-Mills, and in each
case, propose that the full duality group is a nontrivial Z_2 extension of the
duality group acting on bosonic degrees of freedom, to more accurately describe
possible actions on fermions. We also walk through U-duality groups for
toroidal compactifications to nine, eight, and seven dimensions, which enables
us to perform cross-consistency tests of these proposals.
