  We construct a Fock model of the minimal representation of the exceptional
Lie supergroup $\mathbb{D}(2,1, \alpha)$. Explicit expressions for the action
are given by integrating to group level a Fock model of the Lie superalgebra
$D(2,1, \alpha)$ constructed earlier by the authors. It is also shown that the
representation is superunitary in the sense of de Goursac--Michel.
