Universal deformation rings for the symmetric group S_5 and one of its double covers

Let $S_5$ denote the symmetric group on 5 letters, and let ${\widehat{S}}_5$ denote a non-trivial double cover of $S_5$ whose Sylow 2-subgroups are generalized quaternion. We determine the universal deformation rings $R(S_5,V)$ and $R({\widehat{S}}_5,V)$ for each mod 2 representation $V$ of $S_5$ that belongs to the principal 2-modular block of $S_5$ and whose stable endomorphism ring is given by scalars when it is inflated to ${\widehat{S}}_5$. 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 $S_5$ and ${\widehat{S}}_5$, respectively, has an affirmative answer.