Let $F$ be a finite group and $\psi \colon F \to F$ a group automorphism.

Question 1: I'm interested in enumerating/characterizing the finite-index subgroups of $F \rtimes_\psi \mathbb{Z}$ in a 'closed' way, similar to the description that Goursat's lemma provides for the subgroups of $F \times \mathbb{Z}$ (which relies solely on the subgroup structure of $F$). I know that the general problem of enumerating or characterizing subgroups of any semidirect product $F_1 \rtimes F_2$ is somewhat hopeless, but I hope there exists a 'reasonable' description in $F \rtimes_\psi \mathbb{Z}$ in terms of $F$ and $\psi$.

There's this paper that describes the subgroups of any semidirect product $G \rtimes_\phi H$, but I couldn't specialize the results to the case I'm interested in. This question asks for the general case.

Question 2: same as Question 1 but for finite-index subgroups of $G_1 \ast_F G_2$, where $G_1, G_2$ are finite groups such that $[G_1:F] = [G_2:F] = 2$. This paper describes the subgroups of free product with amalgamation, but (again) I couldn't extract any 'down to earth' results. I'm aware that when $T = \{1\}$ (that is, $G$ is the infinite dihedral group) the enumeration is straightforward.

The motivation for this question is to compute the exponential zeta function for the subgroup growth sequence, that is, the series \begin{equation*}g(z) = \exp\left( \sum_{n \geq 0} \frac{a_n(G)}{n}z^n \right)\end{equation*} where $a_n(G)$ is the number of subgroups of index $n$ in $G$ (here $G$ is of the form $F \times \mathbb{Z}$ or $G_1 \ast_T G_2$ as before). Any observations or results for a subclass of these groups are welcome.