ALIBABA 2021 Problem 16

Let $G$ be a finite group, and let $H_1, H_2 \subset G$ be two subgroups. Suppose that for any representation of $G$ on a finite-dimensional complex vector space $V$ , one has that $$\text{dim} V^{H_1}=\text{dim} V^{H_2},$$ where $V^{H_i}$ is the subspace of $H_i$ -invariant vectors in $V$ ( $i=1,2$ ). Prove that $$Z(G) \cap H_1=Z(G) \cap H_2.$$ Here $Z(G)$ denotes the center of $G$ .