= Solution
{c}
Realize the cube's vertices as $(\pm1,\pm1,\pm1)$ and take the four with coordinate product $+1$. They form a <regular tetrahedron>, and their negatives form the other one. Every proper rotation of the first tetrahedron also permutes the negatives, hence is a cube rotation. This embeds $G_1$ as a <subgroup> of $G_2$.
The embedded <group> preserves each of these two four-vertex sets and acts transitively on each. Thus the resulting action on all eight cube vertices has \b[two orbits, each of size four].
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