gr.group theory - Connections between properties of a group and local symmetries of its Cayley graph

I think there is very little that can be said, even when $Gamma$ is finite. Obviously
any automorphism of $Gamma$ that fixes the connection set of the Cayley graph gives
rise to an automorphism of the ball of radius $n$ about $e$; hence if the ball is
asymmetric then there is no automorphism of $Gamma$ that fixes the connection set.
(There can be automorphisms of the Cayley graph that do not arise from automorphisms of the group, but I cannot see that these are relevant to your question.)

To get a result of the type you are asking about, you would need to assume that $2n$
is larger than the girth (or your ball would be a "regular" tree, and tell you nothing
about the group). But now the balls of radius $n$ will be complex structures, and it
is getting difficult to even formulate a result.

The balls do provide information about the spectrum of the adjacency operator, but it
is not clear how to relate this to the group structure in general.

In the finite case there is a lot of work devoted to studying automorphism groups of Cayley
graphs, but I am not aware of any results of the type you describe.