We introduce ${varphi}$-frames in $L^2$(G), as a generalization of a-frames defined in [8], where G is a locally compact Abelian group and ${varphi}$ is a topological automorphism on G. We give a characterization of ${varphi}$-frames with regard to usual frames in $L^2$(G) and show that ${varphi}$-frames share several useful properties with frames. We define the associated ${varphi}$-analysis and ${varphi}$-preframe operators, with which we obtain criteria for a sequence to be a ${varphi}$-frame or a ${varphi}$-Bessel sequence. We also define ${varphi}$-Riesz bases in $L^2$(G) and establish equivalent conditions for a sequence in $L^2$(G) to be a ${varphi}$-Riesz basis.