Commutator map is homomorphism if commutator is in centralizer

Statement
Suppose $$G$$ is a group, $$H$$ is a subgroup of $$G$$, and $$x \in G$$ is such that:

$$[x,H] \subseteq C_G(H)$$

Then, the map $$H \to G$$ given by:

$$y \mapsto [x,y]$$

is a homomorphism of groups from $$H$$ to $$G$$.

Related facts

 * Class two implies commutator map is endomorphism is a special case where the statement applies globally.

Facts used

 * 1) uses::Formula for commutator of element and product of two elements

Proof
The proof follows directly from Fact (1).