The word 'based' refers to the the fairy tales, not to what you are being asked to do.
The phrase '...a fairy tale based on..' is a reduced relative clause. The full form is '...a fairy tale which is based on..' .
The verb form '..is based on..' is in the passive voice, which is why we use the past participle.
If you want to use the present participle to refer to what the people are asked to do while working in groups, you could say, for example:
'Work in groups to make up your own fairy tale, basing it on a familiar tale. '
I hope that makes sense.