  In the article we settle down the problem of permanence of property RD under
group extensions. We show that if $1\to N\to G\to Q\to 1$ is a short exact
sequence of compactly generated groups such that $Q$ has property RD, and $N$
has property RD with respect to the restriction of a word-length on $G$, then
$G$ has property RD.
  We also generalize the result of Ji and Schweitzer stating that locally
compact groups with property RD are unimodular. Namely, we show that any
automorphism of a locally compact group with property RD which distorts
distances subexponentially, preserves the Haar measure.
