Group corresponding to an alternating bilinear map from one odd-order abelian group to another

Statement
Suppose $$K,H$$ are odd-order abelian groups and $$c:K \times K \to H$$ is an alternating bihomomorphism. The corresponding group to this map is a group $$G$$ defined as $$H \times K$$ as set, with multiplication:

$$\! (h,k)(h',k') = (hh', c(h,h')^{1/2}kk')$$