  We investigate how coarse embeddability of box spaces into Hilbert space
behaves under group extensions. In particular, we prove a result which implies
that a semidirect product of a finitely generated free group by a finitely
generated residually finite amenable group has a box space which coarsely
embeds into Hilbert space. This provides a new class of examples of metric
spaces with bounded geometry which coarsely embed into Hilbert space but do not
have property A, generalising the example of Arzhantseva, Guentner and Spakula.
