Number of automorphism for clique of 6 vertices minus two non adjacent edges
It's not hard to check that $(1\; 2)$, $(1\;3)(2\;4)$, and $(5\;6)$ are all automorphisms of the graph. Considering the degrees of vertices, this shows that the orbits of the automorphism group $G$ are $\{1,2,3,4\}$ and $\{5,6\}$. According to the orbit stabilizer theorem, $|G| = 4|G_1|$. Once again, it's not hard to check that $(3\;4)$ and $(5\;6)$ are automorphisms in $G_1$, and consider degrees and the property of being connected to $1$ shows that the orbits of $G_1$ are $\{3,4\}$ and $\{5,6\}$. It is not hard to check that all 4 permutations respecting these orbits are in $G_1$, and so $|G_1| = 4$. We conclude that $|G| = 16$.

Informally, $1$ can get mapped to any of $1,2,3,4$. Assuming that $1$ gets mapped to itself (without loss of generality), we see that $2$ has to get mapped to itself, $3$ to one of $3,4$, and $5$ to one of $5,6$. These are the only constraints, so in total there are $4\cdot 2\cdot 2 = 16$ automorphisms.