  We consider a $D=4$, $\mathcal{N}=2$ gauged supergravity with an
electromagentic Fayet-Iliopoulos term. We restrict to the uncharged, single
dilaton consistent truncation and point out that the bulk Lagrangian is
self-dual under electromagnetic duality. Within this truncation, we construct
two families of exact hairy black hole solutions, which are asymptotically
$AdS_4$. When a duality transformation is applied on these solutions, they are
mapped to two other inequivalent families of hairy black hole solutions. The
mixed boundary conditions of the scalar field correspond to adding a
triple-trace operator to the dual field theory action. We also show that the
this truncation contains all the consistent single dilaton truncations of
gauged $\mathcal{N}=8$ supergravity with a possible $\omega$-deformation.
