  Some classification results for closed surfaces in Berger spheres are
presented. On the one hand, a Willmore functional for isometrically immersed
surfaces into an homogeneous space $\mathbb{E}^{3}(\kappa,\tau)$ with isometry
group of dimension $4$ is defined and its first variational formula is
computed. Then, we characterize Clifford and Hopf tori as the only Willmore
surfaces satifying a sharp Simons-type integral inequality. On the other hand,
we also obtain some integral inequalities for closed surfaces with constant
extrinsic curvature in $\mathbb{E}^3(\kappa,\tau)$, becoming equalities if and
only if the surface is a Hopf torus in a Berger sphere.
