A (para)topological group is simply sm-factorizable exactly when every continuous real-valued function on it factors through a strongly submetrizable group, and for Hausdorff topological groups in this class the Raikov, Dieudonné, and Hewitt-Nachbin completions agree.
Engelking, General Topology, Heldermann, Berlin (1989)
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Simply $sm$-factorizable (para)topological groups and their completions
A (para)topological group is simply sm-factorizable exactly when every continuous real-valued function on it factors through a strongly submetrizable group, and for Hausdorff topological groups in this class the Raikov, Dieudonné, and Hewitt-Nachbin completions agree.