Let $C$ be a smooth projective curve over $\mathbb{C}$ with automorphism group $G$. When is it possible to construct an analytic family of smooth projective curves $\mathcal{C} \longrightarrow T$ (e.g. $T = \mathbb{C}$) such that $\text{Aut }C_t \cong G$ for (almost) every $t \in T$ ($C_t$ is the fiber over $t \in T$)?

I came across a paper that claimed that this is always possible when $G$ is a cyclic group of order $p$ for some odd prime $p$, but I'm not sure what kind of construction the author had in mind or whether this can be generalized. Also, what happens if we replace the condition $\text{Aut }C_t \cong G$ by $|\text{Aut }C_t| \ge G$?