The Schreier conjecture asserts that $\mathrm{Out}(G)$ 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 $M_{11}$, $M_{23}$, $M_{24}$ are sporadic simple groups, whose outer automorphism group is trivial.

Mathieu group $M_{12}$, $M_{22}$ are sporadic simple groups, whose outer automorphism group is $\mathbb{Z}_2$.

Tits group ${}^2F_4(2)′$ is a sporadic simple group, whose outer automorphism group is $\mathbb{Z}_2$.

Monster group $\mathbb{M}$ 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 $\mathbb{Z}_n$?)