Enumerating finite-index subgroups of two-ended groups

Question

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.

Answer 1

A geometric approach to this problem is given by Bass-Serre theory. Here is the sketch of an argument:

Let $H \leq G_1 \ast_F G_2$ be a finite-index subgroup. The action of $H$ on the Bass-Serre tree $T$ (which is a line) of $G_1 \ast_F G_2$ must be cocompact, and the quotient graph provides a decomposition of $H$ as a graph of groups $\mathcal{G}$. Essentially, two cases can happen:

• The underlying graph of $\mathcal{G}$ is a segment. In other words, $H$ acts on $T$ with a segment $S$ as a strict fundamental domain. Vertex-stabilisers (in $H$) of interior vertices of $S$ have to fix $S$ entirely. So $H$ is generated by the stabilisers of the endpoints of $S$. In other words, $H$ is the amalgamated product $A_1 \ast_B A_2$ generated by two non-trivial subgroups $A_1,A_2$ in two distinct conjugates of $G_1$ or $G_2$ (not included in the corresponding copy of $F$).
• The underlying graph of $\mathcal{G}$ is a circle. Then the segment $S$ is not a strict fundamental domain: its endpoints lie in the same orbit. In this case, $H$ is the HNN extension generated by a subgroup $A$ in a conjugate of $G_1$ or $G_2$ (namely, the stabiliser of the first endpoint of $S$) with an infinite-order element $t$ (namely, an element that sends the first endpoint of $S$ to the second).

For $F \rtimes_\psi \mathbb{Z}$, one can argue similarly because it is also an HNN extension of $F$.