If $G$ is a group, we can say $g$ is _automorphism-conjugate_ to $f$ if there is a group automorphism $\alpha : G \to G$ such that $g = \alpha(f)$. This is an equivalence relation.

 > Is there a standard name for this equivalence relation? Is it a well-studied notion? Does it have some importance somewhere?

Some alternative ways to state this:
 * like conjugacy classes, we can talk about automorphism orbits in  group, and I am just asking for the name of / literature on the related equivalence, or equivalence relation.
 * $f, g \in G$ are automorphism-conjugate if and only if $f$ and $g$ are actually conjugate in the holomorph of $G$.

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.