The Schreier conjecture asserts that Out(G) is always a solvable group when G is a finite simple group. This result is known to be true as a corollary of the classification of finite simple groups, although no simpler proof is known.

Question:

• What is the obstruction to have a direct transparent proof of Schreier conjecture?

Mathieu group M11, M23, M24 are sporadic simple groups, whose outer automorphism group is trivial.

Mathieu group M12, M22 are sporadic simple groups, whose outer automorphism group is $\mathbb{Z}_2$.

Monster group M is the largest sporadic simple group, whose outer automorphism group is trivial.

• Are there some bounds for the outer automorphism groups of sporadic simple groups? (such as bounded by a finite cyclic group $\mathbb{Z}_n$?)