Generalized Dedekind symbols for modular forms of real weights

Fukuhara defined generalized Dedekind symbols as functions
on P^1(Q) with values in an abelian group satisfying a short list of
relations.
In a previous paper,  I have generalized this definition to the case
of possibly non-commutative groups and
constructed non--commutative generalized Dedekind symbols
for classical  PSL(2,Z ) cusp forms, using iterated period polynomials.
Here I generalize this construction to forms of real weights using their
iterated period functions introduced and studied in a recent  article 
by R.~Bruggeman and Y.~Choie.