I am in a situation where a discrete, finitely generated group $H$ satisfies property (T), and was wondering if I was able to conclude anything about the pair $(G,H^G)$, where $G$ is a finitely generated discrete group containing $H$, and $H^G$ is the normal closure of $H$ in $G$. In particular, does this pair necessarily also satisfy (T)? If not, are there conditions which would guarantee that it did?

Thanks for your time!