  We introduce a notion of cocycle-induction for strong uniform approximate
lattices in locally compact second countable groups and use it to relate
(relative) Kazhdan- and Haagerup-type of approximate lattices to the
corresponding properties of the ambient locally compact groups. Our approach
applies to large classes of uniform approximate lattices (though not all of
them) and is flexible enough to cover the $L^p$-versions of Property (FH) and
a-(FH)-menability as well as quasified versions thereof a la Burger--Monod and
Ozawa.
