  We study the behaviour on some nodal hyperplanes of the isomorphism,
described in a paper of 2019 by Boissi\`ere, Camere and Sarti, between the
moduli space of smooth cubic threefolds and the moduli space of hyperk\"ahler
fourfolds of $K3^{[2]}$-type with a non-symplectic automorphism of order three,
whose invariant lattice has rank one and is generated by a class of square 6;
along those hyperplanes the automorphism degenerates by jumping to another
family. We generalize their result to singular nodal cubic threefolds having
one singularity of type $A_i$ for $i=2, 3, 4$ providing birational maps between
the loci of cubic threefolds where a generic element has an isolated
singularity of the types $A_i$ and some moduli spaces of hyperk\"ahler
fourfolds of $K3^{[2]}$-type with non-symplectic automorphism of order three
belonging to different families. In order to treat the $A_2$ case, we introduce
the notion of K\"ahler cone sections of $K$-type generalizing the definition of
$K$-general polarized hyperk\"ahler manifolds.
