• Trivially, non-abelian simple groups and the full group of automorphisms are almost simple. For ${\displaystyle n=5}$  or ${\displaystyle n\geq 7,}$  the symmetric group ${\displaystyle \mathrm {S} _{n}}$  is the automorphism group of the simple alternating group ${\displaystyle \mathrm {A} _{n},}$  so ${\displaystyle \mathrm {S} _{n}}$  is almost simple in this trivial sense.
• For ${\displaystyle n=6}$  there is a proper example, as ${\displaystyle \mathrm {S} _{6}}$  sits properly between the simple ${\displaystyle \mathrm {A} _{6}}$  and ${\displaystyle \operatorname {Aut} (\mathrm {A} _{6}),}$  due to the exceptional outer automorphism of ${\displaystyle \mathrm {A} _{6}.}$  Two other groups, the Mathieu group ${\displaystyle \mathrm {M} _{10}}$  and the projective general linear group ${\displaystyle \operatorname {PGL} _{2}(9)}$  also sit properly between ${\displaystyle \mathrm {A} _{6}}$  and ${\displaystyle \operatorname {Aut} (\mathrm {A} _{6}).}$ 

The full automorphism group of a non-abelian simple group is a complete group (the conjugation map is an isomorphism to the automorphism group),^[2] but proper subgroups of the full automorphism group need not be complete.

By the Schreier conjecture, now generally accepted as a corollary of the classification of finite simple groups, the outer automorphism group of a finite simple group is a solvable group. Thus a finite almost simple group is an extension of a solvable group by a simple group.

• Quasisimple group
• Semisimple group

• Almost simple group at the Group Properties wiki