Which Mobius transformations, explicitly, are you talking about? The general set of Mobius functions is infinite, you must have a particular 6 in mind....I suspect you mean:

f(x) = x

g(x) = 1/x (this is an involution)

h(x) = 1 - x (this is also clearly an involution)

k(x) = x/(x - 1) (this is not so obviously an involution)

m(x) = 1/(1 - x)

n(x) = (x - 1)/x

to construct an isomorphism with S_{3}, follow the hint: do the involutions fix one of {0,1,∞}? Which functions correspond to 3-cycles?