Let me denote the graph in the picture by $\Gamma$ and I will refer to the points as numbers $1-9$. I need to find Aut($\Gamma$).
Looking at this graph, it seems that there can be a permutation on the outer most edges 17 and 96, then both, then a symmetric mirror of this graph. So we have id,(17),(96),(17)(96),(25)(16)(97),(25)(19)(67),(25)(16)(97)(34),(25)(19)(67)(34). Is this correct? are there any other automorphisms I'm missing? thanks.