If $G$ is a finite group then ... Let $G$ be a finite group. Let $I(G)$ be the set of elements of $G$ that have order 2. Suppose $|I(G)| \ge \frac 3 4 |G|$. Let $x \in I(G)$. We note $I_x(G)$ the subset $\{xg \in I(G)\}_{g \in I(G)}$. Show that $|I_x(G)| \ge \frac 1 2 |G|$.
I have shown that $I_x(G) \subset C_G(x)$ but I have no idea how to continue from there.. Can somebody please give me a hint (I'm not asking for the answer, I'd like to find that myself).

You are looking at the intersection of two sets, both of which have size $\ge \frac{3}{4}|G|$.
