Isomorphism of semidirect products of surface groups

Question

Recall that the fundamental group of a closed Riemann surface of genus $h$ has the presentation $$\Pi_h= \langle a_1, \,b_1, \ldots, a_h,\, b_h \; | \; [a_1, \, b_1]\ldots [a_h, \, b_h]=1 \rangle.$$ Assume now that we have two split short exact sequences of groups \begin{equation*} \begin{split} & 1 \longrightarrow \Pi_g \longrightarrow G_1 \longrightarrow \Pi_b \longrightarrow 1 \\ & 1 \longrightarrow \Pi_g \longrightarrow G_2 \longrightarrow \Pi_b \longrightarrow 1 \\ \end{split} \end{equation*}

Question. Is there a theoretical or computational way to check whether $G_1$ and $G_2$ are isomorphic or not?

Note that I'm talking about abstract isomorphism of the middle groups, not of isomorphism of sequences. In my specific situation (where $g=41$ and $b=2$), I know the conjugacy action of $\Pi_b$ on $\Pi_g$ in both cases, so I can obtain explicit semidirect-type presentations for $G_1$ and $G_2$ and I can feed them to GAP4.

In this way, I can check that $G_1$ and $G_2$ have the same abelianization. But the problem about their isomorphism type eludes me for the moment.

Every answer or reference to the relevant literature will be greatly appreciated.

Answer 1

If the action of $\Pi_b$ on $\Pi_g$ via conjugation in $G$ has non-trivial kernel and infinite image in the outer automorphism group $Out(\Pi_g)$ then the extension structure for $G$ is unique. If the image of $\Pi_b$ in $Out(\Pi_g)$ is finite then $G$ is virtually a product, and there are at most two such extension structures. (This is clear when $G\cong\Pi_g\times\Pi_b$ is a product!) Otherwise, if the action is injective there are finitely many such extension structures.

These results are in (or flow from) ``A group-theoretic analogue of the Parshin-Arakelov Theorem", by F. E. A. Johnson, Archiv Math. (Basel) 63 (1994), 354-361.

Let $d=(2-2g)(2-2b)$ be the Euler characteristic of $G$. Johnson's work implies an upper bound of the order of $d^d$ on the number of such extension structures when the action is injective. On the other hand, Nick Salter showed that the number of such extension structures can be exponential in $d$.