I don't understand the nerve construction. For $\mathbb{Z}_2$, Wikipedia says $\bullet \overset{1}\longrightarrow \bullet \overset{1}\longrightarrow \bullet$ should produce a nondegenerate 2-simplex, but I don't see how if there's only one vertex/object and two edges/arrows - by face maps or inspection, the edge $\bullet \overset{1}\longrightarrow \bullet$ is repeated. Some edge must be repeated in any 2-simplex, so all 2-simplices should be degenerate.
So the problem in particular is the geometric realization of two loops on a point is a figure 8, which should be the classifying space of $\mathbb{Z}_2$, doesn't have fundamental group $\mathbb{Z}_2$. And that's stretching it, because the edges are degenerate, so if it's a true complex it should just be a point, with trivial fundamental group.