(For sporadic groups – i.e. those not falling in an infinite family – the notion of exceptional outer automorphism is ill-defined, as there is no general formula.)
To see that S6 has an outer automorphism, recall that homomorphisms from a group G to a symmetric group Sn are essentially the same as actions of G on a set of n elements, and the subgroup fixing a point is then a subgroup of index at most n in G. Conversely if we have a subgroup of index n in G, the action on the cosets gives a transitive action of G on n points, and therefore a homomorphism to Sn.
The fact that it is possible to construct this automorphism at all relies on a large number of numerical coincidences which apply only to n = 6.
Identifying PGL(2, 5) with S5 and the projective special linear group PSL(2, 5) with A5 yields the desired exotic maps S5 → S6 and A5 → A6.
S5 acts transitively on the coset space, which is a set of 120/20 = 6 elements (or by conjugation, which yields the action above).
Ernst Witt found a copy of Aut(S6) in the Mathieu group M12 (a subgroup T isomorphic to S6 and an element σ that normalizes T and acts by outer automorphism).
The most visual way to see this automorphism is to give an interpretation via algebraic geometry over finite fields, as follows.
The restriction of q to H has defect line L, so there is an induced quadratic form Q on the 4-dimensional H/L that one checks is non-degenerate and non-split.
The zero scheme of Q in H/L defines a smooth quadric surface X in the associated projective 3-space over k. Over an algebraic closure of k, X is a product of two projective lines, so by a descent argument X is the Weil restriction to k of the projective line over a quadratic étale algebra K. Since Q is not split over k, an auxiliary argument with special orthogonal groups over k forces K to be a field (rather than a product of two copies of k).
The natural S6-action on everything in sight defines a map from S6 to the k-automorphism group of X, which is the semi-direct product G of PGL2(K) = PGL2(9) against the Galois involution.
To see that none of the other symmetric groups have outer automorphisms, it is easiest to proceed in two steps: The latter can be shown in two ways: Each permutation of order two (called an involution) is a product of k > 0 disjoint transpositions, so that it has cyclic structure 2k1n−2k.
If one forms the product of two distinct transpositions τ1 and τ2, then one always obtains either a 3-cycle or a permutation of type 221n−4, so the order of the produced element is either 2 or 3.
On the other hand, if one forms the product of two distinct involutions σ1, σ2 of type k > 1, then provided n ≥ 7, it is always possible to produce an element of order 6, 7 or 4, as follows.
Any representative of the outer automorphism constructed above exchanges the conjugacy classes, whereas an index 2 subgroup stabilizes the transpositions.