Almost simple not implies simple or complete

Statement

There can exist a finite almost simple group that is neither a simple group nor a complete group.

In other words, there can exist a finite simple non-Abelian group $S$ that is not complete, and also such that $S$ is not a maximal normal subgroup of $\operatorname {Aut} (S)$. In other words, there exist intermediate normal subgroups between $S$ and $\operatorname {Aut} (S)$.

Proof

Example of the alternating group of degree six

Further information: Alternating group:A6, symmetric group:S6

Let $S$ be the alternating group of degree six and $G$ be the symmetric group of degree six. The automorphism group of $S$ is a group containing $G$ with index two, namely, a semidirect product of $G$ by an outer automorphism of order two. Thus, $S$ is not a maximal subgroup of $\operatorname {Aut} (S)$ -- $G$ is an intermediate subgroup.

Example of projective special linear groups

If $q$ is a power of an odd prime such that $4|q-1$, then the group $PSL(4,q)$ has index four inside $PGL(4,q)$, and there is an intermediate subgroup. Since $PGL(4,q)$ is itself contained in the automorphism group of the simple group $PSL(4,q)$, the intermediate subgroup is a subgroup that is neither equal to the simple group nor equal to the whole automorphism group.