  In this note we revisit a less known symmetrization method for functions with
respect to a topological group $G$, which we call $G$-averaging. We note that,
although quite non-technical in nature, this method yields $G$-invariant
minimizers of functionals satisfying some relaxed convexity properties. We give
an abstract theorem and show how it can be applied to the $p$-Laplace and
polyharmonic Poisson problem in order to construct symmetric solutions. We also
pose some open problems and explore further possibilities where the method of
$G$-averaging could be applied to.
