I'm wondering if it is possible to define the following kind of scheme:

Suppose X is a scheme and G is a subgroup of the endomorphism group of

X. Define for each x in X the orbit of x under G

[x] := { gx : g in G }.

Now define the set of orbits of X under G

Y := { [x] : x in X }.

My question is this: Is there a natural way of giving Y a structure of

scheme using the scheme structure of X?

If this is already a well-studied topic, can someone please point me to

where I may look this up?

Thanks.

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-kira