= Solution
For a real vector <relativistic quantum field>, $V^\mu=V^{\mu\dagger}$. Its transformed field must also be a <Hermitian operator>, so the <intrinsic charge-conjugation phase> satisfies $\eta_C=\eta_C^*$; combined with $|\eta_C|=1$, this gives
$$
\boxed{\eta_C=\pm1.}
$$
The <quantum electrodynamics> interaction is $-e\bar\psi\gamma^\mu\psi A_\mu$. Since the <Dirac electromagnetic current> is odd under <charge conjugation>, <charge conjugation> invariance of this interaction requires the <photon> field to be odd too:
$$
\boxed{\eta_C(A)=-1,\qquad \hat C A^\mu\hat C^{-1}=-A^\mu.}
$$
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