  The concept of formal duality was proposed by Cohn, Kumar and Sch\"urmann,
which reflects a remarkable symmetry among energy-minimizing periodic
configurations. This formal duality was later on translated into a purely
combinatorial property by Cohn, Kumar, Reiher and Sch\"urmann, where the
corresponding combinatorial objects were called formally dual pairs. Almost all
known examples of primitive formally dual pairs satisfy that the two subsets
have the same size. Indeed, prior to this work, there was only one known
example having subsets with unequal sizes in $\mathbb{Z}_2 \times
\mathbb{Z}_4^2$. Motivated by this example, we propose a lifting construction
framework and a recursive construction framework, which generate new primitive
formally dual pairs from known ones. As an application, for $m \ge 2$, we
obtain $m+1$ pairwise inequivalent primitive formally dual pairs in
$\mathbb{Z}_2 \times \mathbb{Z}_4^{2m}$, which have subsets with unequal sizes.
