  On the set $\mathcal M$ of mean functions the symmetric mean of $M$ with
respect to mean $M_0$ can be defined in several ways. The first one is related
to the group structure on $\mathcal M$ and the second one is defined trough
Gauss' functional equation. In this paper we provide an answer to the open
question formulated by B.\ Farhi about the matching of these two different
mappings called symmetries on the set of mean functions. Using techniques of
asymptotic expansions developed by T.\ Buri\'c, N.\ Elezovi\'c and L.\
Mihokovi\'c (Vuk\v si\'c) we discuss some properties of such symmetries trough
connection with asymptotic expansions of means involved. As a result of
coefficient comparison, new class of means was discovered which interpolates
between harmonic, geometric and arithmetic mean.
