Let $G$ be a finite group, and suppose we have a complex representation $V$ of $\mathbb C$. Let $\langle , \rangle$ be an arbitrary (Hermitian) inner product on the $\mathbb C$-vector space $V$, and define $(,) : V \times V \to \mathbb C$ by $(x,y) = \displaystyle \frac{1}{|G|} \sum_{g \in G} \langle gx, gy \rangle$. Then $(,)$ is a Hermitian inner product. Pick $h \in G$. Then (hx, hy) = \displaystyle \frac{1}{|G|} \sum_{g \in G} \langle ghx, ghy \rangle = \frac{1}{|G|} \sum_{g' \in G} \langle g'x,g'y \rangle = (x,y).
The point of this is to show that all $\mathbb C$-representations of a finite group $G$ have a $G$-invariant Hermitian inner product. I'm being a bit slow here: why are we dividing by $|G|$?