Group actions on filtered modules and finite determinacy. Finding large submodules in the orbit by linearization

Let M be a module over a local ring R and a group action G M, not necessarily R-linear. To understand how large is the G-orbit of an element z ∈ M one looks for the large submodules of M lying in Gz. We provide the corresponding (necessary/sufficient) conditions in terms of the tangent space to the orbit, T(Gz,z).
This question originates from the classical finite determinacy problem of Singularity Theory. Our treatment is rather general, in particular we extend the classical criteria of Mather (and many others) to a broad class of rings, modules and group actions.
When a particular ‘deformation space’ is prescribed, Σ ⊆ M, the determinacy question is translated into the properties of the tangent spaces, T(Gz,z), T(Σ,z), and in particular to the annihilator of their quotient, annT(Σ,z)/T(Gz,z).