Consider the simple undirected graph $G$ with natural equivalence relation $\sim$ on $V(G)$:

$u\sim v$ iff they are similar, i.e. iff there exists $\phi\in Aut(G)$ with $\phi(u)=v$.

Define a "quotient graph" $G_{Aut}$ in the following way:

$V(G_{Aut})=V(G)/\sim$ and there is an edge $A-B$ iff $\exists \ a\in A, b\in B$ with $ab\in E(G)$.

**Conjecture**: If in $G$ every pair of similar vertices are non-adjacent, then $G_{Aut}\subset G \ ?$

ADDED:As Anton showed, this conjecture is false. But what one can said if $G$ is a tree? Does conjecture remains false?

truefor trees (I proved it just now). I can post the outline of the proof if you're interested [or email it to you or something]. It's not difficult to prove anyway. – M. Vinay Oct 8 '15 at 9:04