= Solution
Start with any positive-definite Hermitian inner product $\langle-,-\rangle$ on $\mathbb C^n$ and average it over the finite group:
$$
\langle v,w\rangle_G=
\frac1{|G|}\sum_{g\in G}
\langle\rho(g)v,\rho(g)w\rangle.
$$
This remains positive definite. For $h\in G$, reindexing $g\mapsto gh$ gives
$$
\langle\rho(h)v,\rho(h)w\rangle_G
=\langle v,w\rangle_G,
$$
so the form is $G$-invariant.
Choose an orthonormal basis for $\langle-,-\rangle_G$. In this basis every $\rho(g)$ preserves the standard Hermitian form and is therefore unitary. If $P$ is the change-of-basis matrix, then
$$
\rho'(g)=P^{-1}\rho(g)P
$$
defines an isomorphic representation with $\rho'(G)\leq U_n$.
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