I have a puzzle about Schroedinger equation with time-dependent hamiltonian, which is usually used in time-dependent quantum systems.

However, one of the axioms in quantum mechanics postulates that the hamiltonian as the generator of the one-parameter unitary group U(t) is not time-dependent, and the evolution of a quantum state should obey the Schroedinger equation with time-independent hamiltonian.

So why can Schroedinger equation with time-dependent hamiltonian be used without hesitation?

Where does the time-dependent hamiltonian come from, except for the case of interaction picture? And should there be an additional axiom for it?