Is there a well-understood correspondence between subgroups G of $SL_2(\mathbb{Z})$ (not necessarily of finite index) and graded algebras of modular forms invariant under G?

Given an algebra of modular forms (on the upper half-plane) known to be an algebra of forms M(G) invariant under some modular subgroup G can one reconstruct the subgroup G? (Possibly, the group G should arise as a deck transformation group for the covering Proj M(G) -> Proj M($SL_2(\mathbb{Z}))=(?)\mathbb{P}^1(\mathbb{C})$)) arising from the natural embedding M($SL_2(\mathbb{Z})$)) -> M(G) .)

(By the way, what is a good reference for general properties of Proj M(G)?)

If this is not enough can we reconstruct the (modular) subgroup from the knowledge of its invariant algebras of p-adic modular forms?

What can we say about the situation in case of discrete subgroups of $PSL_2(\mathbb{R})$?

Do you know any papers considering the situation of fuchsian groups (structure of algebra of modular forms etc.)?

Do you know any papers considering algebras of classical and p-adic modular forms for the case of subgroups G of infinite index in $SL_2(\mathbb{Z})$?