In Galois theory, we are to discuss about splitting field $\mathbb{Q_f}$ of a polynomial $f\in \mathbb{Q}[x]$ and build correspondence between the galois subgroup $H\leq Gal(\mathbb{Q_f/Q})$ and a subextension fixed field $\mathbb{Q}\leq K\leq\mathbb{Q_f}$, that $Gal(\mathbb{Q_f}/K)\cong H$.
Could this similar idea be applied to graph theory? For instance, if we consider a graph $G$ and its automorphism subgroup $H\lhd Aut(G)$, is there somehow a corresponding graph, say, a "quotient graph" $G/H$ defined by its group action orbit, and having that $Aut(G/H)\cong Aut(G)/H$?