Locally inner not implies inner

This article gives the statement and possibly, proof, of a non-implication relation between two automorphism properties. That is, it states that every automorphism satisfying the first automorphism property (i.e., locally inner automorphism) need not satisfy the second automorphism property (i.e., inner automorphism)
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Statement

It is possible to have a group $G$ and an automorphism $\sigma$ of $G$ such that $\sigma$ is a locally inner automorphism of $G$ (i.e., the restriction of $\sigma$ to every finitely generated subgroup of $G$ is inner on that subgroup) but $\sigma$ is not an inner automorphism of $G$.

Facts used

1. Locally inner automorphism-balanced not implies central factor

Proof

The result follows from Fact (1). Note that the group $G$ for the statement we are trying to prove here will be the subgroup $H$ in the language on the page of Fact (1), and the automorphism $\sigma$ will be an automorphism arising by restricting an inner automorphism of the ambient group that does not restrict to an inner automorphism.

Instead of using Fact (1) as a black box, we could also use either of the examples for it explicitly:

• The case that $G$ is the finitary symmetric group and $\sigma$ is conjugation by an infinitary permutation.
• The case that $G$ is a restricted direct product of centerless groups and $\sigma$ is conjugation by an element of the unrestricted direct product that has infinitely many non-identity coordinates.