Self-centralizing and minimal normal implies fully invariant in co-Hopfian group

Statement
Suppose $$G$$ is a fact about::co-Hopfian group and $$H$$ is a fact about::minimal normal subgroup of $$G$$ that is also a fact about::self-centralizing subgroup, i.e., $$C_G(H) \le H$$. Then, $$H$$ is a fact about::fully invariant subgroup of $$G$$.

Facts used

 * 1) uses::Self-centralizing and minimal normal implies monolith
 * 2) uses::Monolith is fully invariant in co-Hopfian group

Proof
The proof follows directly from facts (1) and (2).