 Solution

Realize the cube's vertices as $(\pm 1,\pm 1,\pm 1)$ 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 two orbits, each of size four.