= Solution
An <automorphism of a root system> outside the <Weyl group> is the sign change
$$
\boxed{\tau(\varepsilon_1)=\varepsilon_1,\qquad
\tau(\varepsilon_2)=\varepsilon_2,\qquad
\tau(\varepsilon_3)=-\varepsilon_3.}
$$
It preserves the set $\{\pm\varepsilon_i\pm\varepsilon_j\}$, fixes $\alpha_1$, and interchanges $\alpha_2$ and $\alpha_3$. It therefore realizes the nontrivial symmetry of the <Dynkin diagram> while preserving the chosen <root basis>.
To verify that $\tau\notin W$, note that $s_1$ swaps $\varepsilon_1,\varepsilon_2$, $s_2$ swaps $\varepsilon_2,\varepsilon_3$, and $s_3$ swaps $\varepsilon_2,\varepsilon_3$ while changing both their signs. Every product of these generators is a signed permutation with an even number of sign changes, whereas $\tau$ has one sign change. Thus it cannot be a <Weyl group> element. The parity here concerns sign changes, rather than the determinant of the permutation.
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