Question related to binomial distribution

Question

Let $X \sim Binomial (n, p)$ with both $n$ and $p$ known. Suppose for some non-increasing function $G:[0,1] \rightarrow [0,1]$, and some fixed $c_0 \in [0,1]$, we have that \begin{align} \label{ineq} P(G(X/n) \leq c_0) \leq \alpha \end{align} where $\alpha$ can be chosen as 0.1. Let $\mathcal{G}$ be set that contains all the functions satisfy the above inequality. I want to find the smallest such function $G_0$ such that $G_0 \in \mathcal{G}$ and $G_0(x) \leq G(x), \quad \forall x \in [0,1]$, where $G$ is any other function in $\mathcal{G}$. Can anyone give some hint on looking for such a function. Thank you very much. Hannah