  The manifold which admits a genus-$2$ reducible Heegaard splitting is one of
the $3$-sphere, $\mathbb{S}^2 \times \mathbb{S}^1$, lens spaces and their
connected sums. For each of those manifolds except most lens spaces, the
mapping class group of the genus-$2$ splitting was shown to be finitely
presented. In this work, we study the remaining generic lens spaces, and show
that the mapping class group of the genus-$2$ Heegaard splitting is finitely
presented for any lens space by giving its explicit presentation. As an
application, we show that the fundamental groups of the spaces of the genus-$2$
Heegaard splittings of lens spaces are all finitely presented.
