Let H be a normal subgroup of G and let a belong to G. If the element aH has order 3 in the group G/H and H has order 10, what are the possibilities for the order of a?

a^3 is an element of H, an element of H has what possible orders?

so the possible orders would be 1, 2, 5, 10?

the possible orders of elements of H are indeed 1,2,5,10, so a^3 has possible orders 1,2,5,10 and hence a has possible order..?

3, 6, 15, 30?

Yep.