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We investigate how coarse embeddability of box spaces into Hilbert spacebehaves under group extensions. In particular, we prove a result which impliesthat a semidirect product of a finitely generated free group by a finitelygenerated residually finite amenable group has a box space which coarselyembeds into Hilbert space. This provides a new class of examples of metricspaces with bounded geometry which coarsely embed into Hilbert space but do nothave property A, generalising the example of Arzhantseva, Guentner and Spakula.

Box spaces, group extensions and coarse embeddings into Hilbert space