On the embedding problem for $2^{+}{S}_4$ representations

Let $2^{+}{S}_4$ denote the double cover of $S_4$ corresponding to the element in $H^2(S_4,\mathbb{Z}\mathrm{/}2\mathbb{Z})$ where transpositions lift to elements of order 2 and the product of two disjoint transpositions to elements of order 4 (denoted $\tilde S_4$in\text{cite}Serre)\mathrm{.}Givenanellipticcurve$E$,let$E[2]$ denote its 2-torsion points. Under some conditions on $E$ (as in \cite{Bayer}) elements in $H^1({\text{Gal}}_{\text{Q}},E[2])\mathrm{\backslash }\{0\}$ correspond to Galois extensions $N$ of $\text{Q}$ with Galois group (isomorphic to) $S_4$. On this work we give an interpretation of the addition law on such fields, and prove that the obstruction for $N$ having a Galois extension $\tilde{N}$ with $\Gal(\tilde N/ \Q) \simeq 2^+S_4$givesanhomomorphism$s_4^+:H^1(\Gal_\Q,E[2]) \to H^2(\Gal_\Q,\Z/2\Z)$\mathrm{.}AsaCorollarywecanprove(if$E$ has conductor divisible by few primes and high rank) the existence of 1$-dimensional representations attached to $E$ and use them in some examples to construct 3/2 modular forms mapping via the Shimura map to (the modular form attached to) $E$.