Graph Automorphisms and Induced Subgraphs Let $G=(V,E)$ be a finite simple graph, $\Gamma=Aut(G)$ be the automorphism group of $G$, and $G_v=G-\{v\}$ be a vertex-deleted induced subgraph of $G$. We define the following equivalence relation on the vertices,
$u,v\in V: u\sim v \iff G_u\cong G_v$
(I will forego proving that this is an equivalence as it seems trivial. If someone would like further clarification please let me know.) This equivalence partitions $V$ such that,
$V=\displaystyle\bigcup_{i=1}^k V_i$
We know that $u\sim v$ is a necessary condition for $u$ and $v$ to lie in the same orbit under $Aut(G)$ (i.e. these equivalence classes are larger than the orbits of $V$ under $\Gamma$).

Can we recover some information about $\Gamma$ given the equivalence classes under $\sim$?

It is clear that a partition of $V$ into orbits under $\Gamma$ will be a refinement of the partition. Also, we have an obvious upper bound $\Gamma\le S_{|V|}$. It seems that the "easier case" is when there are many vertices with distinct equivalence classes (i.e. the graph is not very symmetric).

The multiset of isomorphisms types (equivalence classes in your question) induced by the removal of every single vertex is called the deck of $G$. Certainly, "some information" can be recovered from the deck. For example, in a number of special cases (classes of graphs), we know that $G$ can be reconstructed unambiguously from its deck. Thus, a fortiori, $\Gamma$ as well (from $G$). Whether this is true in general is not known and is referred to as the reconstruction conjecture [1].
In fact, the reconstruction conjecture is at least as strong as your question, since the reconstruction of $G$ yields that of $\Gamma$. Whether the converse holds is not clear to me (can we determine $G$ from $\Gamma$ and from the deck?). For sure, some information can be recovered from the deck that helps ruling out some automorphisms, e.g. the degree of every vertex and a few other such parameters are known to be determined unambiguously by the deck.
[1] https://en.wikipedia.org/wiki/Reconstruction_conjecture
