Obviously, Aut(G) X Aut(H) is a subgroup of Aut(G\vee H). However, if the graphs are isomorphic, then you can swap corresponding vertices between the graphs. You get additional swaps when: G' is a subgraph of G, H' is a subgraph of H, G' is isomorphic to H', every vertex in G' is adjacent to every vertex in V(G-G'), and every vertex in H' is adjacent to every vertex in V(H-H'). Are there other possibilities for an automorphism on the join?
Vitaly Roman'kov
If $G$ is a union of graphs $G_1, ..., G_l$ that are pairwise disconnected, and $H$ is a similar union of $H_1, ..., H_t$ we can swap some pairs of isomorphic $G_i$ and $H_j$. Some mixed automorphisms using the author's ideas and these posibility are possible too.
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