Why does $L^1(G)$ characterize a locally compact group?

Question

In a book I am reading it is stated that if $G_1,G_2$ are locally compact groups such that their convolution algebras $L^1(G_1),L^1(G_2)$ are isometrically isomorphic, then $G_1$ and $G_2$ are topologically isomorphic. However, the book provides no proof or reference for this statement. Is there an elementary proof of this fact? If not, then a reference to a proof would be greatly appreciated.