  We consider the family of complex algebraic K3 surfaces $\mathcal{X}$ with
Picard lattice containing the unimodular lattice $H \oplus E_7(-1) \oplus
E_7(-1)$. The surface $\mathcal{X}$ admits a birational model isomorphic to a
quartic hypersurface that generalizes the Inose quartic. We prove that a
general member of this family admits exactly four inequivalent Jacobian
elliptic fibrations and construct explicit pencils for them.
