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

Statement

Suppose $G$ is a Co-Hopfian group (?) and $H$ is a Minimal normal subgroup (?) of $G$ that is also a Self-centralizing subgroup (?), i.e., $C_{G}(H)\leq H$. Then, $H$ is a Fully invariant subgroup (?) of $G$.

Facts used

1. Self-centralizing and minimal normal implies monolith
2. Monolith is fully invariant in co-Hopfian group

Proof

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