Suppose I have a group $G$, and I have a group automorphism $\alpha : G \to G$ such that $g = \alpha(f)$. In case $\alpha$ is inner this is standard conjugacy. In general it's some kind of natural notion of similarity.

Is there a standard name? Is it a well-studied notion? Does it have some importance somewhere? I feel like I have encountered this before, but I am not even able to find a name in the literature (I found it a little hard to search for because Google confuses it with e.g. conjugacy in automorphism groups, and a range of other things that sound vaguely similar).

I'm thinking about this because I realized that if $G$ is a big homeomorphism group, Rubin's theorem shows that automorphism-conjugacy is a notion between group-theoretic and topological conjugacy (because group automorphisms have to come from topological conjugacies), and these two notions are of great interest to me.