  The unitary extension principle (UEP) by Ron and Shen yields conditions for
the construction of a multi-generated tight wavelet frame for $L^2(\mr^s)$
based on a given refinable function. In this paper we show that the UEP can be
generalized to locally compact abelian groups. In the general setting, the
resulting frames are generated by modulates of a collection of functions, via
the Fourier transform this corresponds to a generalized shift-invariant system.
Both the stationary and the nonstationary case are covered. We provide general
constructions, based on B-splines on the group itself as well as on
characteristic functions on the dual group. Finally, we consider a number of
concrete groups and derive explicit constructions of the resulting frames.
