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.