  We prove the existence of a projective good moduli space of principal
$\mathcal{G}$-bundles under nonconnected reductive group schemes $\mathcal{G}$
over a smooth projective curve $C$. We also prove that the moduli stack of
$\mathcal{G}$-bundles decomposes into finitely many substacks
$\text{Bun}_{\mathcal{P}}$ each admitting a torsor
$\text{Bun}_{\mathcal{G}_\mathcal{P}}\to \text{Bun}_{\mathcal{P}}$ under a
finite group, for some connected reductive group schemes
$\mathcal{G}_\mathcal{P}$ over $C$. We use this for the second purpose of the
article: for any constant connected reductive group $G$, the subset of
essentially finite $G$-bundles in the moduli of degree 0 semistable $G$-bundles
over $C$ is not dense, unless $G$ is a torus or the genus of $C$ is smaller
than 2. We do this by giving an upper bound on the dimension of the closure of
the subset of essentially finite $G$-bundles.
