This is an exercise, 7.1 of the book "An Introduction to the group theory by J.J.Rotman":
Which of the following properties, when enjoyed by both $K$ and $Q$, is also enjoyed by every extension of $K$ by $Q$?
i. solvable
ii. nilpotent
iii. periodic
iv. torsion-free
I know that if $H$ and $G/H$ is solvable so is $G$, then i. is correct. ii. is not correct because $D_{\infty}$ is an extension of $\mathbb Z$ by $ \mathbb Z_2$ but it is not nilpotent. Diving in the groups makes my mind distracted and couldn't find a proper counter-example about iii. and iv. Thanks for you help.