= Solution
A function element is a pair $(f,D)$ consisting of a domain $D\subseteq G$ and a <holomorphic function> $f:D\to\mathbb C$ belonging to the given complete analytic function. Its <germ of a holomorphic function> at $z\in D$ is the equivalence class $[f]_z$, where two elements are equivalent when they agree on a neighbourhood of $z$.
For every function element $(f,D)$ and sufficiently small open $U\subseteq D$, set
$$
U_f=\{[f]_z:z\in U\}.
$$
These sets form a basis for the <space of germs of holomorphic functions>. The projection
$$
\pi:\mathcal G\to G,
\qquad \pi([f]_z)=z,
$$
restricts to a homeomorphism $U_f\to U$. The inverse projections $\pi|_{U_f}$ are compatible complex charts, making $\mathcal G$ a <Riemann surface> and $\pi$ a covering map and local biholomorphism. In such a chart, the <evaluation map on a space of germs> has coordinate expression
$$
\mathcal E\circ(\pi|_{U_f})^{-1}(z)=f(z).
$$
It is therefore holomorphic on every chart, hence analytic on each component of $\mathcal G$.
Now let $f:R\to S$ be a nonconstant analytic map of compact connected Riemann surfaces, and let $B$ be its finite branch-value set. Each point of $B$ is a <branch value of a holomorphic map>. At every point of $R\setminus f^{-1}(B)$ the local degree is one, so the holomorphic inverse-function theorem supplies a neighbourhood on which $f$ is biholomorphic. For $y\in S\setminus B$, the fibre is finite. Choose disjoint inverse neighbourhoods around all its points and shrink their common image; compactness, or equivalently properness of $f$, ensures that no additional preimages enter. Thus
$$
f:R\setminus f^{-1}(B)\longrightarrow S\setminus B
$$
is an ordinary unramified covering map.
Fix $P\in S\setminus B$. Given a loop $\gamma$ based at $P$, lift it from every $Q\in f^{-1}(P)$ and send $Q$ to the endpoint of its lift. The <path lifting theorem> makes this a permutation; concatenation of loops composes the permutations and reversal gives inverses. Its image is the <monodromy group of a covering>. The punctured surface $R\setminus f^{-1}(B)$ is path connected. Given two points of the fibre, join them there by a path; its projection is a loop whose lift joins those points. Hence the monodromy action is transitive.
Finally consider
$$
F(z)=z^2+z^{-2}=\frac{z^4+1}{z^2}.
$$
It has degree four. Its finite critical points satisfy
$$
F'(z)=2z-2z^{-3}=0,
\qquad z^4=1,
$$
with branch values $2$ and $-2$; the double poles at zero and infinity give the branch value infinity. For a regular value, the fibre is
$$
\{z,-z,z^{-1},-z^{-1}\}.
$$
The two commuting involutions $z\mapsto-z$ and $z\mapsto z^{-1}$ preserve $F$ and act transitively on this fibre. They generate all four deck transformations, so the covering is normal and its monodromy group is
$$
\boxed{\ H\cong V_4=C_2\times C_2.\ }
$$
With the displayed ordering of the fibre, its elements are
$$
1,\qquad(12)(34),\qquad(13)(24),\qquad(14)(23).
$$
This is the <monodromy group of z squared plus z to the minus two>.
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