One common way to find such a isomorphism would be to use the Cayley Theorem.
My prof. also mentioned that there are many other isomorphism, such as mapping the generators for D2 to (1,2) and (3,4) respectively or mapping the generators for D2 to (1,3) and (2,4) respectively, where the mappings are defined as f(ab) = f(a)f(b).
How did he derive such isomorphisms?
Edit: D2 refers to the dihedral group of order 4. S4 refers to the permutation group on a set of 4 objects.