Here an almost simple group is a finite group whose socle (product of all minimal normal subgroups) is a nonabelian simple group. As an extension of its socle, an almost simple group could be split or non-split. For example, there are four groups with socle $L=PSL(2,9)$ other than $L$, i.e. $S\_6$, $PGL(2,9)$, $M\_{10}$ and $P\Gamma L(2,9)$; the former two are split extensions of $L$ while the latter two are not. Now the question is **how to determine all the almost simple groups which is a non-split extension of its socle?** 



When is an almost simple group a split extension of its socle?
See the following paper:
A. Lucchini, F. Menegazzo, M. Morigi.
On the existence of a complement for a finite simple group in its automorphism group. Special issue in honor of Reinhold Baer (1902–1979). 
Illinois J. Math. 47 (2003), no. 1-2, 395–418. MR2031330


