  Fix a finite group $G$. We study $\Omega^{SO,G}_2$ and $\Omega^{U,G}_2$, the
unitary and oriented bordism groups of smooth $G$-equivariant compact surfaces,
respectively, and we calculate them explicitly. Their ranks are determined by
the possible representations around fixed points, while their torsion subgroups
are isomorphic to the direct sum of the Bogomolov multipliers of the Weyl
groups of representatives of conjugacy classes of all subgroups of $G$. We
present an alternative proof of the fact that surfaces with free actions which
induce non-trivial elements in the Bogomolov multiplier of the group cannot
equivariantly bound. This result permits us to show that the 2-dimensional
SK-groups (Schneiden und Kleben, or ``cut and paste") of the classifying spaces
of a finite group can be understood in terms of the bordism group of free
equivariant surfaces modulo the ones that bound arbitrary actions.
