Semiconjugacy class relative to an endomorphism

Definition
Let $$G$$ be a group and $$\alpha:G \to G$$ be an endomorphism. A semiconjugacy class with respect to $$\alpha$$ is an equivalence class under the following relation:

$$x \sim y \iff \exists g \in G, gx\alpha(g)^{-1} = y$$.

That the relation is an equivalence relation is easily verified.

In the case that $$\alpha$$ is the identity map, a semiconjugacy class is the same thing as a conjugacy class.