  For a Reproducing Kernel Hilbert Space on a complex domain we give a formula
that describes the Hermitean metrics on the domain which are pull-backs of some
metric on the (dual of) the RKHS via the evaluation map. Then we consider the
question when such metrics are invariant with respect to the group of
automorphisms of the domain. First we approach the problem by considering a
stronger property, demanding that the original metric on the (dual of) the RKHS
is invariant with respect to all (adjoints of) composition operators, induced
by automorphisms. However, we show that only the trivial metric satisfies this
property. Then we characterise RKHS's for which the Bergman metric analogue
studied in \cite{cd} and \cite{arsw} is automorphism-invariant.
