  In this work, we introduce the ${\mathbb Z}_3$-graded differential algebra,
denoted by $\Omega(\widetilde{\rm GL}_q(2))$, treated as the ${\mathbb
Z}_3$-graded quantum de Rham complex of ${\mathbb Z}_3$-graded quantum group
$\widetilde{\rm GL}_q(2)$. In this sense, we construct left-covariant
differential calculi on the ${\mathbb Z}_3$-graded quantum group
$\widetilde{\rm GL}_q(2)$.
