  Let $S_5$ denote the symmetric group on 5 letters, and let $\hat{S}_5$ 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
$R(\hat{S}_5,V)$ 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 $\hat{S}_5$. 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
$\hat{S}_5$, respectively, has an affirmative answer.
