  Let $G$ be a locally compact group with an open subgroup $H$ with the
Kunze-Stein property, and let $\pi$ be a unitary representation of $H$. We show
that the representation $\widetilde{\pi}$ of $G$ induced from $\pi$ is an
$L^{p+}$-representation if and only if $\pi$ is an $L^{p+}$-representation. We
deduce the following consequence for a large natural class of almost
automorphism groups $G$ of trees: For every $p \in (2,\infty)$, the group $G$
has a unitary $L^{p+}$-representation that is not an $L^{q+}$-representation
for any $q < p$. This in particular applies to the Neretin groups.
