Powering-invariance is centralizer-closed

Statement
Suppose $$G$$ is a group and $$H$$ is a powering-invariant subgroup of $$G$$. Then, the centralizer of $$H$$ in $$G$$, i.e., the group $$C_G(H)$$, is also a powering-invariant subgroup of $$G$$.

Facts used

 * 1) uses::c-closed implies powering-invariant

Proof
By Fact (1), the centralizer of any subgroup in a group is powering-invariant. In particular, the centralizer of a powering-invariant subgroup is powering-invariant.