Finite holonomy acts on a small transverse disk

(Domains before invariance.) In transverse coordinates with $x=0$, choose one representative $f_h$ of each of the finitely many germs, with $f_1=\mathrm{i}\mathrm{d}$. There are neighborhoods $U_1⋐U_0$ of zero such that every $f_h$ is defined on $U_0$, every $f_h(U_1)$ lies in $U_0$, and $f_g\circ f_h=f_{gh}$ on $U_1$ for every $g,h\in H$. Indeed each relation is a germ equality, and only finitely many domains, images and relations have to be accommodated. No invariance of $U_1$ is assumed.

(Invariant neighborhood.) Put $D_0={\bigcap }_{h\in H}{f}_h(U_1)$. This open neighborhood of zero lies in $U_1$ because $f_1=\mathrm{i}\mathrm{d}$. For $z\in D_0$ and each $h$, write $z=f_h(y_h)$ with $y_h\in U_1$. Then $f_g(z)=f_{gh}(y_h)\in f_{gh}(U_1)$ for every $h$, so $f_g(D_0)\subseteq D_0$. The inverse relation on $D_0$ gives equality. Hence these restrictions realize a genuine action. Work in the connected component containing zero, which every $f_h$ preserves.

(A disk and extensions.) Let $A_h=D{f}_h(0)$; F4 gives $A_{gh}=A_g{A}_h$. On $D_0$ set $k(z)=\mathrm{\mid }H{\mathrm{\mid }}^{-1}{\sum }_h{A}_h^{-1}{f}_h(z)$. Then $Dk(0)=I$ and reindexing the sum gives $k(f_g(z))=A_{g}k(z)$. F4 gives a smooth inverse for $k$ near zero. Shrink that inverse domain by intersecting its finitely many group translates, so it remains an invariant neighborhood on which $k$ is injective. Average a Euclidean inner product over the linear maps $A_h$. A sufficiently small ball for that inner product, with its closure inside the image of the inverse domain, is invariant under every $A_h$. Its inverse image $D$ under $k$ is therefore an invariant open disk with compact closure inside $D_0\subseteq U_1$. Every $f_h$ is defined on $U_0$, a neighborhood of that closure. In transverse dimension zero the same assertions hold with $D=\{x\}$.

(Prescribed metric case.) If a smooth Riemannian metric germ is supplied, choose the representatives and domains of step 1.1 inside its common isometry domain. On the connected $D_0$ the resulting action is by isometries fixing $x$, so it preserves intrinsic distance from $x$. By F5 a sufficiently small such metric ball is a normal exponential ball and hence an open disk: take its radius below the first-exit bound for a relatively compact normal neighborhood. Its compact closure lies in $D_0$, so the extensions from step 1.1 still apply. This uses the supplied metric, without replacing it by an unrelated averaged one.

Thus finite holonomy is represented by a smooth action on an invariant transverse disk, with each representative defined past its closure. In the prescribed Riemannian metric case this disk may be chosen to be a metric ball.