A back tuck requires a more conditioned body than a double full. It is a very important basic skill. To perform a back tuck you must first sit into the athletic stance. Then swing arms to ears, and then begin your extension upwards into a stretch jump. As you stretch upwards your arms lift you (that is their job).
Once you have achieved your stretch jump, you pull your shins and toes up in a karate kick type style over your head rotating in a candlestick tuck position. It is the same position that you end up in after a backward roll. As your knees, shins and toes come over your head you may do one of two different things with your arms. Your arms can stay by your ears until your body rotates up to meet them or you can complete the circle with arms and grab under your thighs to help your rotation. At no time do you bring your arms forward after the stretch jump. That changes your backward momentum into forward momentum, which is bad news for a back tuck.