Prove $A_4$ acts on the set of edges of the tetrahedron T and find the stabilizer of $A_{4_{(1,2)}}$

Question

I'm trying to solve the following task:

I don't know how im supposed to go about this. How can the set of even permutations of 4 elements act on a 6-element set.

I understand it can permute the 4 points, and that reflects on the edge-set. But would that be acting on the Vertex set, and not the edge set? The edge set differences would change only as a consequence.

I'm really having difficulties with both a) and b). Please help me solve this.

Thank you in advance.