= Solution
Denote the three stated pair sums by $r_1,r_2,r_3$. They are pairwise distinct: for example,
$$
r_1-r_2=(\alpha_1-\alpha_4)(\alpha_2-\alpha_3)\ne0,
$$
and the other differences factor similarly. Every <Galois automorphism> fixes each $r_i\in F$, and hence fixes each of the three corresponding partitions of four roots into two pairs. The kernel of the permutation action $S_4\to S_3$ on these partitions is
$$
V_4=\{1,(12)(34),(13)(24),(14)(23)\}.
$$
Thus $G$ embeds in the <Klein four-group>. By part (a), $4$ divides $|G|$, and the containing group has order four. Consequently
$$
\boxed{G\cong V_4}.
$$
This <Split resolvent criterion for a Klein four Galois group> uses the <cubic resolvent of a quartic>. The pair-sum differences remain nonzero in characteristic two, so this proof does not need the missing characteristic assumption from part (c).
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