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Let $S_5$ denote the symmetric group on 5 letters, and let $\hat{S}_5$ denotea non-trivial double cover of $S_5$ whose Sylow 2-subgroups are generalizedquaternion. 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 theprincipal 2-modular block of $S_5$ and whose stable endomorphism ring is givenby scalars when it is inflated to $\hat{S}_5$. We show that for these $V$, aquestion raised by the first author and Chinburg concerning the relation of theuniversal deformation ring of $V$ to the Sylow 2-subgroups of $S_5$ and$\hat{S}_5$, respectively, has an affirmative answer.

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