  Ultraviolet fixed point functions of the functional renormalisation group
equation for $f(R)$-gravity coupled to matter fields are discussed. The metric
is split via the exponential parameterisation into a background and a
fluctuating metric, the former is chosen to be the one of a four-sphere. Also
when scalar, fermion and vector fields are included global quadratic solutions
exist as in the pure gravity case for discrete sets of values for some
endomorphism parameters defining the coarse-graining scheme. The asymptotic,
large-curvature behaviour of the fixed point functions is analysed for generic
values of these parameters. Examples for global numerical solutions are
provided. A special focus is given to the question whether matter fields might
destabilise the ultraviolet fixed point function. Similar to a previous
analysis of a polynomial, small-curvature approximation to the fixed point
functions different classes for such functions are found.
