Universal deformation rings for the symmetric group S5 and one of its double covers

Let S5 denote the symmetric group on 5 letters, and let denote a non-trivial double cover of S5 whose Sylow 2-subgroups are generalized quaternion groups. We determine the universal deformation rings R(S5,V) and of each mod 2 representation V of S5 that belongs to the principal 2-modular block of S5 and whose stable endomorphism ring is given by scalars when it is inflated to . We show that for these V, a question raised by the first author and Chinburg concerning the relation of the universal deformation ring of V to the Sylow 2-subgroups of S5 and , respectively, has an affirmative answer.