Finitely generated inner automorphism group implies every locally inner automorphism is inner

Statement
Suppose $$G$$ is a group with finitely generated inner automorphism group, i.e., the inner automorphism group $$\operatorname{Inn}(G)$$ (which is isomorphic to the quotient group $$G/Z(G)$$ of $$G$$ by its center $$Z(G)$$) is a finitely generated group. Then, $$G$$ is a group in which every locally inner automorphism is inner. In other words, for every locally inner automorphism $$\sigma$$ of $$G$$ (i.e., $$\sigma$$ resembles an inner automorphism on any finite subset of $$G$$), $$\sigma$$ is an inner automorphism of $$G$$.

Proof idea

 * First show that $$\sigma$$ is a center-fixing automorphism, i.e., it sends every element of the center to itself.
 * Pick a finite generating set of $$G/Z(G)$$ and choose a finite subset of $$G$$ that maps to this set under the quotient map. The inner automorphism that $$\sigma$$ resembles on this finite subset must in fact equal $$\sigma$$ on all of $$G$$.

Proof details
Given: A group $$G$$ such that $$G/Z(G)$$ is finitely generated, a locally inner automorphism $$\sigma$$ of $$G$$. Here, $$Z(G)$$ denotes the center of $$G$$.

To prove: There exists $$g \in G$$ such that $$\sigma(x) = gxg^{-1}$$ for all $$x \in G$$.

Proof: