I ran across this paper recently, which explores Conway's proof that $L_4(2) = A_8$: https://archive.maths.nuim.ie/staff/jmurray/Preprints/jmurrayalt8.pdf . A central part of the argument involves the automorphism group of $E16$, the elementary abelian group with 16 elements, and a version of $E16$ built from Cayley embeddings of copies of $E8$ into $S_8$. This is related to the $E16$ of synthemes in the classic Sylvester construction of the outer automorphism of $S_6$, and used to extend the embedding of $S_6$ into $L_4(2)$ to an embedding of $A_8$ into $L_4(2)$; a group-size argument then proves these are the same group.
This argument is very interesting to me because it exhibits a construction of $A_8$ as the automorphism group of $E16$. I have a vague idea that embedding $S_6$ in $A_8$ is a step on the way to connecting its exotic outer automorphism to the triality phenomenon in $SO(8)$ and friends. So I went looking for an automorphism-group construction connected a bit more closely to $SO(8)$, which I sometimes think of in terms of the Clifford algebra $Cl_{0,6}(\mathbb{R})$. At the level of an associative algebra, $Cl_{0,6}(\mathbb{R})$ is isomorphic to $M_8(\mathbb{R})$, with the 2-forms spanning $so(8)$, and the signed monomials form a finite group under the Clifford product isomorphic to $2_{+}^{1+6}$, the extraspecial group of order 128 and type + (which is SmallGroup(128,2326) in GAP). The signed monomials of even grade form a subgroup that can be described as a central extension of $2_{+}^{1+4}$ (or $2_{-}^{1+4}$) by the cyclic group $C_4$, in which the central $C_4$ contains the scalar and top form (pseudoscalar). Quotienting this even subgroup (which is SmallGroup(64,266) in GAP; see comments by https://math.stackexchange.com/users/2820/derek-holt at Is this extension of $Sp(4,2)$ a semidirect product?) by its center yields $E16$. It has an automorphism group which, like $A_8$, is an extension of $S_6$; GAP reports its structure as "(C2 x C2 x C2 x C2 x C2) : S6".
I'm moderately confident that the automorphism group of $S_6$ is a subgroup of this group; but I'm new to GAP (and in general haven't looked at finite groups much in a long time) and have a lot to learn in order to explore this further. I also suspect that this is not quite the right object to be focusing on if I want to connect up with triality; more likely I should be looking at the $2_{-}^{1+6}$ (which is SmallGroup(128,2327) in GAP) in $Cl_{6,0}(\mathbb{R})$. (And I should probably be looking not just at the exotic outer automorphism but the exotic Schur cover of $A_6$?)
I'd like to learn more about automorphisms of "groups with auxiliary structure", such as these monomial groups within (the process of constructing) Clifford algebras. Which subgroup contains the monomials of even grade depends on the signature of the Clifford algebra, and presumably some automorphisms of the group as a whole preserve the even subgroup and others don't — and that's more group theory than I ever studied formally. Perhaps someone can recommend a reference or three? And perhaps specifically someone could explain whether there's a structure than contains and unifies both the exotic Schur cover of $A_6$ and its exotic automorphism group. Does Wilson cover this in https://webspace.maths.qmul.ac.uk/r.a.wilson/fsgs.html ?