Suppose I wish to find the set of isomorphism classes of $\mathbb{Z}/n\mathbb{Z}$-equivariant line bundles over a 2-dimensional, compact $\mathbb{Z}/n\mathbb{Z}$-CW-complex $X$, i.e. $\mathrm{Vect}^{1}_{\mathbb{Z}/n\mathbb{Z}}(X)$, I can write it using equivariant homotopy theory as

 $\mathrm{Vect}^{1}_{\mathbb{Z}/n\mathbb{Z}}(X)\approx[X,B_{\mathbb{Z}/n\mathbb{Z}}GL_{1}(\mathbb{C})]_{\mathbb{Z}/n\mathbb{Z}}$.

Nonequivariantly, $B_{\mathbb{Z}/n\mathbb{Z}}GL_{1}(\mathbb{C})$ is $K(\mathbb{Z},2)$ so the above would seem like a Bredon cohomology group $H^2_{\mathbb{Z}/n\mathbb{Z}}(X;M)$ with $M$ some coefficient system and I could try to use the universal coefficient spectral sequence to attempt at computing it, however it seems like an awful lot of machinery for one of the simpler cases.

If the action is relatively nice, though it has fixed points, is there a simpler way of finding
$\mathrm{Vect}^{1}_{\mathbb{Z}/n\mathbb{Z}}(X)$? 

If not, which coefficient system $M$ is the right one?

Thanks for your time