$\newcommand{\cD}{\mathcal{D}}\newcommand{\cX}{\mathcal{X}}$(This is a slight rephrasing and modification of the original question)
Let $D\subset\mathbb{C}$ be the complex unit disk. Let $X$ be a compact complex manifold, fibered over $D$ in relative dimension 1. Assume the fibers above $D^* := D - 0$ are smooth, and the fiber above 0 has at worst ordinary double points (nodes) as singularities. For any $\eta\in D^*$, associated to each node $x\in X_0$ is a vanishing cycle $v_x\in H_1(X_\eta)$, and the Picard-Lefschetz formula describes precisely the monodromy action of a generator $\gamma\in\pi_1(D^*,\eta)$ on $H_1(X_\eta)$ in terms of the vanishing cycles $v_x$. In particular, if $\varphi$ is the monodromy representation, then $\varphi(\gamma)$ is unipotent.
Let $\mu_n$ be a cyclic group of order $n$ acting by rotations on $D$. Suppose that this action on $D$ lifts to an action on $X$ (in particular it acts on $X_0$). Let $X^* := X - X_0$, then $X^*/\mu_n\rightarrow D^*/\mu_n$ is a family of smooth curves over a punctured disk, and if $\eta'$ denotes the image of $\eta$, then we may again ask about the monodromy action of a generator $\gamma'\in\pi_1(D^*/\mu_n,\eta')$ on $H_1(X_\eta)$. Abusing notation, again let $\varphi$ denote the monodromy representation. Let $\varphi(\gamma') = SU$ (say, viewed in $\text{GL}(H_1(X_\eta,\mathbb{C}))$ be the Jordan-Chevalley decomposition, so $S$ commutes with $U$, $S$ is semisimple, and $U$ is unipotent.
By comparison with the situation over $D$, we know that $$\varphi(\gamma'^n) = \varphi(\gamma)$$ which forces $S^n = 1$ and $U^n = \varphi(\gamma)$, so $U$ is the unique unipotent $n$th root of $\varphi(\gamma)$.
If $\mu_n$ acts freely on the nodes, then $X/\mu_n$ is again a manifold, so applying Picard-Lefschetz directly to $X/\mu_n$, we find that $\varphi(\gamma')$ is unipotent, so $S = 1$.
My question concerns the case where $\mu_n$ fixes the nodes, and also preserves the branches at every node. In this case, how can we relate the semisimple part $S$ of $\varphi(\gamma')$ to information about how $\mu_n$ acts on $X_0$?
