Extensions and Kazhdan's Property (T) Is Kazhdan's property&nbsp;(T) stable under extensions? i.e. if $G$ is an extension of
a group with property&nbsp;(T) by a group with property&nbsp;(T), does it follow that $G$ has property&nbsp;(T)?
The answer to your question is yes:

See Bekka&ndash;de&nbsp;la&nbsp;Harpe&ndash;Valette, Kazhdan's property $(T)$, page&nbsp;69 (the standard reference on property&nbsp;(T), freely available on Bekka's homepage):



The proof is not very difficult, and it is given in a clear fashion in the book, so it doesn't make much sense to reproduce it here.

Note also that Property&nbsp;(T) is closed under taking quotients, but it is not closed under passing to normal subgroups. The standard example is the semi-direct product $\mathrm{SL}_{n}(\mathbb{R}) \ltimes \mathbb{R}^n$ which has property&nbsp;(T) for $n \geq 3$ but whose normal subgroup $\mathbb{R}^n$ doesn't.
If I understood your question correctly, the answer is NO.  For instance, take property T to be being cyclic.  The Klein-4 group is not cyclic, but it has three subgroups of order 2, all of them cyclic.