  P-E. Caprace and N. Monod isolate the class $\mathscr{X}$ of locally compact
groups for which relatively amenable closed subgroups are amenable. It is
unknown if $\mathscr{X}$ is closed under group extension. In this note, we
exhibit a large, group extension stable subclass of $\mathscr{X}$, which
suggests $\mathscr{X}$ is indeed closed under group extension. Along the way,
we produce generalizations of the class of elementary groups and obtain
information on groups outside $\mathscr{X}$.

Group Theory