I would say that it means I have a lagrangian $\mathcal{L}$ that depends on a bunch of fields. I can transform those fields under $G$. They may or may not transform under the same representation $\rho_G$ of some group $G$.  The objects in any given representation are not invariant under the group (unless they are in a trivial representation). It's the system as a whole. So the system really does have the symmetry $G$, not $\rho_G$. 

Examples:

* In QCD one has SU(3) gauge symmetry. The quarks transform under the fundamental. The gluons transform under the adjoint. The baryons are scalars under SU(3). They are all different representations but the symmetry of the system as a whole is the group symmetry $G=SU(3)$.
* Global Lorentz symmetry. In the Standard model one has scalars (don't transform), fermions (spin 1/2 rep) and vector bosons (spin 1). All of which are in different representations but the whole system has a Lorentz symmetry. 
* Conformal symmetry. One has different conformal weight operators but everything transforms under the same conformal symmetry.

Said in a slightly different way, the system has a symmetry $G$, it has different components all being affected differently by a symmetry transformation in $G$, but at the end of the day, the system is invariant under the action of $G$ as a whole, not any specific representation.