You are wrong but I think not fundamentally: your argument about the order of the
image is right, but you oversaw the fact that the order of any of the three 2-cycles
you wrote there indeed is 2, so you can define , and
the same can be done with any of the other two elements of order 2 in that direct sum.
Of course, the above is not the whole story since you still have
to figure out how to define f on the other elements of that sum...