On automorphisms of groups which extend as automorphisms to every larger group

Solution 1:

How about this: A Characterization of Inner Automorphisms Paul E. Schupp Proceedings of the American Mathematical Society Vol. 101, No. 2 (Oct., 1987), pp. 226-228

https://www.jstor.org/stable/2045986?seq=1#page_scan_tab_contents

Abstract: It turns out that one can characterize inner automorphisms without mentioning either conjugation or specific elements. We prove the following

THEOREM Let $G$ be a group and let $\alpha$ an automorphism of $G$. The automorphism $\alpha$ is an inner automorphism of $G$ if and only if $\alpha$ has the property that whenever $G$ is embedded in a group $H$, then $\alpha$ extends to some automorphism of $H$.