  We study threefolds fibred by Kummer surfaces associated to products of
elliptic curves, that arise as resolved quotients of threefolds fibred by
certain lattice polarized K3 surfaces under a fibrewise Nikulin involution. We
present a general construction for such surfaces, before specializing our
results to study Calabi-Yau threefolds arising as resolved quotients of
threefolds fibred by mirror quartic K3 surfaces. Finally, we give some
geometric properties of the Calabi-Yau threefolds that we have constructed,
including expressions for Hodge numbers.
