Is there a name for $f\mapsto g\circ f\circ g^{-1}$?

Question

Let $H$ and $K$ sets, $f:H\to H$ a map and $g:H\to K$ a bijection.

Then $g$ induces a map $f^g:K\to K$ by $f^g:=g\circ f\circ g^{-1}$. Is there a name for the map $f\to f^g$?

In the special case when $H$ is a group, $K=H$ and both $f$ and $g$ are automorphisms of $H$ then $f\to f^g$ is conjugation of $f$ by $g$. But what is it in general? Induced map? Pushforward? Lifting?

Answer 1

"Induced" is broad enough and it fits. So at the very least it's not wrong to use.