  We study the partition function of the orbifold ABJM theory on $S^3$, which
is the $\mathcal{N}=4$ necklace quiver Chern-Simons-matter theory with
alternating levels, in the Fermi gas formalism. We find that the grand
potential of the orbifold ABJM theory is expressed explicitly in terms of that
of the ABJM theory. As shown previously, the ABJM grand potential consists of
the naive but primary non-oscillatory term and the subsidiary
infinitely-replicated oscillatory terms. We find that the subsidiary
oscillatory terms of the ABJM theory actually give a non-oscillatory primary
term of the orbifold ABJM theory. Also, interestingly, the perturbative part in
the ABJM theory results in a novel instanton contribution in the orbifold
theory. We also present a physical interpretation for the non-perturbative
instanton effects.
