Schreier's conjecture states that the outer automorphism group of any finite simple group is solvable. It has been proved as a consequence of the Classification of Finite Simple Groups, by computing all the outer automorphism groups.
Having looked at the list on the atlas (and knowing that sporadics and alternating groups don't really matter here), I am tempted to say that actually every outer automorphism group of any finite simple group is supersolvable (meaning it has a composition series of normal subgroups with cyclic quotients). Is it true? If not, how many counterexamples are there?
No, the outer automorphism groups are not always supersoluble. An example where this fails is $G = O_8^{+}(3)$, where $\operatorname{Out}(G) \cong S_4$. As Derek points out in the comments section, $O_8^{+}(q)$ (or $D_4(q)$) are, in fact, the only exceptions. Indeed, if $q=p^e$ with $p$ odd, then $\operatorname{Out}(O_8^{+}(q)) \cong S_4 \times C_e$ (see Wilson's "The Finite Simple Groups", bottom of p. $75$).