Reynolds operators  are often given by projecting onto an invariant subspace of a group action.
*The "Reynolds operator" considered by  was essentially the projection of a fluid flow to the "average" fluid flow, which can be thought of as projection to time-invariant flows. Here the group action is given by the action of the group of time-translations.
*Suppose that G is a reductive algebraic group or a compact group, and V is a finite-dimensional representation of G. Then G also acts on the symmetric algebra SV of polynomials.  The Reynolds operator R is the G-invariant projection from SV to the subring SV of elements fixed by G.