Locally inner not implies inner

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) uses::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.