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 $$\operatorname{Aut}(S)$$.

Example of the alternating group of degree six
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.