Want to do the $2$-faces action. We use the Orbit stabilizer theorem. Let $X$ be the set of faces (any face can go to any face), $X=\{1,2,3,4,5,6,7,8 \}$. Where $1,2,3,4$ are the front faces of the picture and the rest are the back faces of the picture.
$|X|=|G:G_1|=|G|/|G_1|$ so $|G|=8|G_1|$ so we need to find $G_1$, which is the stabilizer at $1$.
We can do three symmetries with planes, red to red, purple to purple and blue to blue. And then there is the identity rotation. So far we have $4$ symmetries. What are the last two? Because we need to find $6$ symmetries in total.
There are three rotational symmetries that fix the face $1$; one is the identity, one is $\sigma := (238)(457)$, and the other is $\sigma^2 = \sigma^{-1}$.