  We study the following question: under what conditions extension of one
residually nilpotent group by another residually nilpotent group is residually
nilpotent? We prove some sufficient conditions under which this extension is
residually nilpotent. Also, we study this question for semi-direct products
and, in particular, for extensions of free group by infinite cyclic group: $F_n
\rtimes_{\varphi} \mathbb{Z}$. We find conditions under which this group is
residually nilpotent, find conditions under which this group has long lower
central series. In particular, we prove that for $n=2$ the length of the lower
central series of $F_n \rtimes_{\varphi} \mathbb{Z}$ is equal to 2, $\omega$ or
$\omega^2$.
