  We study the stated skein modules of marked 3-manifolds. We generalize the
splitting homomorphism for stated skein algebras of surfaces to a splitting
homomorphism for stated skein modules of 3-manifolds. We show that there exists
a Chebyshev-Frobenius homomorphism for the stated skein modules of 3-manifolds
which extends the Chebyshev homomorphism of the skein algebras of unmarked
surfaces originally constructed by Bonahon and Wong. Additionally, we show that
the Chebyshev-Frobenius map commutes with the splitting homomorphism. This is
then used to show that in the case of the stated skein algebra of a surface,
the Chebyshev-Frobenius map is the unique extension of the dual Frobenius map
(in the sense of Lusztig) of $\mathcal{O}_{q^2}(SL(2))$ through the triangular
decomposition afforded by an ideal triangulation of the surface. In particular,
this gives a skein theoretic construction of the Hopf dual of Lusztig's
Frobenius homomorphism. A second conceptual framework is given, which shows
that the Chebyshev-Frobenius homomorphism for the stated skein algebra of a
surface is the unique restriction of the Frobenius homomorphism of quantum tori
through the quantum trace map.
