In quantum mechanics the states of a physical system live on a Hilbert space $\mathcal{H}$ and the Hamiltonian operator can be seen as a map $H_0: \mathcal{H} \to \mathcal{H}$. Algebraically this corresponds to an automorphism in the space of states which I will abstractly call $C$ forgetting the necessary for now restriction of the Hilbert space. Thus we view Hamiltonian operators $H_0$ as group elements: $$ H_0 \in Aut(C) $$
More generally, one can consider time varying Hamiltonians $H(t)$. These take the generic form: $$ H(t) = H_0 + \sum_{i\geq 1} c_i H_i(t) $$ where $c_i$ are coefficients valued in a field like $\mathbb{C}$ or such and $H_i$ are functions that contain the "time" information. As a matter of fact, in quantum computing, one considers the time varying part to be a result of some external fields $\epsilon(t)$: $$ H_i(t) = h_i + \sum_{j} \epsilon_{ij}(t)\tilde{h}_{ij}. $$
Question Does the space of time-varying Hamiltonians have a different algebraic structure than that of $Aut(C)$? Or, even better, what could be a good description of the functions $\epsilon(t)$?
The time dependence is a bit confusing for me but that said, I can totally see that for a fixed time $t=t'$, $$H(t)|_{t=t'} \in Aut(C).$$
So what is the best description for $H(t)$ generically as well as for the external fields $\epsilon(t)$?