Automorphism group of a group

Symbol-free definition
The automorphism group of a group is defined as a group whose elements are all the automorphisms of the base group, and where the group operation is composition of automorphisms. In other words, it gets a group structure as a subgroup of the group of all permutations of the group.

Definition with symbols
The automorphism group of a group $$G$$, denoted $$\operatorname{Aut}(G)$$, is a set whose elements are automorphisms $$\sigma:G \to G$$, and where the group multiplication is composition of automorphisms. In other words, its group structure is obtained as a subgroup of $$\operatorname{Sym}(G)$$, the group of all permutations on $$G$$.

Subgroups
Every group-closed automorphism property gives rise to a normal subgroup of the automorphism group. Some of the most important examples are given below:

Facts

 * Extensible equals inner: An automorphism of a group has the property that it can be extended to an automorphism for any bigger group containing it if and only if the automorphism is an inner automorphism.
 * Quotient-pullbackable equals inner: An automorphism of a group has the property that it can be pulled back to an automorphism for any group admitting it as a quotient, if and only if the automorphism is an inner automorphism.

Group families
For various group families, the automorphism group can be described in terms of parameters for members of the families. The descriptions are sometimes quite complicated, so we simply provide links:

Grouping by order
We give below the information for the group cohomology (and hence in particular, the Schur multipliers) for groups of small orders: