I'm trying to solve the following exercise.
We have a vertex-transitive directed graph $G$ and $H\leq G$. $A$ is a collection of all subdigraphs of $G$ isomorphic to $H$.
$D = \{V(C): C\in A \} $
Prove that if $D$ is a partition of $V(G)$ then $D$ is an $\operatorname{Aut}(G)$-invariant partiton.
My thoughts about this:
Since $D$ is a partition of $V(G)$ we know that $D$ is a set of all blocks conjugate to $B$, where $B\subseteq V(G)$ is a block of $G$ whenever $g\in G :g(B)\cap B= \emptyset$ or $B$.
If it's also an $\operatorname{Aut}(G)$-invariant partition then the set D should also be a set of all the $g'\in \operatorname{Aut}(G) : g'(B)\cap B= \emptyset$ or $B$.
Now I don't know how to prove this implication. I would kindly ask for your help :)