  Given a group $G$ and an integer $n\geq2$ we construct a new group
$\tilde{{\cal K}}(G,n)$. Although this construction naturally occurs in the
context of finding new invariants for complex algebraic surfaces, it is related
to the theory of central extensions and the Schur multiplier. A surprising
application is that Abelian groups of odd order possess naturally defined
covers that can be computed from a given cover by a kind of warped Baer sum.
