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 $\operatorname{H}^2(S_4,\mathbb{Z}/2\mathbb{Z})$ where transpositions lift to elements of order $2$ and the product of two disjoint transpositions to elements of order $4$. Given an elliptic curve $E$, let $E[2]$ denote its $2$-torsion points. Under some conditions on $E$ elements in $\operatorname{H}^1(\operatorname{Gal}_{\mathbb{Q}},E[2])\backslash \{ 0 \}$ correspond to Galois extensions $N$ of $\mathbb{Q}$ with Galois group (isomorphic to) $S_4$. In 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 $\operatorname{Gal}(\tilde N/ \Q) \simeq 2^+S_4$ gives a homomorphism $s_4^+:\operatorname{H}^1(\operatorname{Gal}_{\mathbb{Q}},E[2]) \rightarrow \operatorname{H}^2(\operatorname{Gal}_\mathbb{Q}, \mathbb{Z}/2\mathbb{Z})$. As a corollary we can prove (if $E$ has conductor divisible by few primes and high rank) the existence of $2$-dimensional representations of the absolute Galois group of $\mathbb{Q}$ attached to $E$ and use them in some examples to construct $3/2$ modular forms mapping via the Shimura map to (the modular form of weight $2$ attached to) $E$.