Linearly extensible automorphism

Origin
The notion arose through attempts to use representation theory in solving the Extensible automorphisms problem.

Symbol-free definition
Given a group and a field, an automorphism of the group is termed linearly extensible if it extends to an inner automorphism (of the general linear group)for every faithful finite-dimensional linear representation of the group over the field.

Definition with symbols
Given a group $$G$$ and a field $$k$$ an automorphism $$\sigma$$ of $$G$$ is termed linearly extensible if for any faithful linear representation $$\varphi: G \to GL_n(k)$$, there exists $$a \in GL_n(k)$$ such that for any $$g \in G$$:

$$\varphi(\sigma(g)) = a\varphi(g)a^{-1}$$

Stronger properties (subject to further conditions)

 * Weaker than::Inner automorphism (unconditionally)
 * Finite-extensible automorphism when the field is a prime field and the group is a finite group.
 * Class-preserving automorphism when the field is class-determining for the group, which happens to be true in the non-modular case
 * Galois-class automorphism in the non-modular case
 * Weaker than::Linearly pushforwardable automorphism

Weaker properties (subject to further conditions)

 * Class-preserving automorphism when the field is class-separating (for instance, when the group is finite and the field is a splitting field for the group)
 * Galois-class automorphism in the non-modular case