Do you want to move the grouped layers on the page, or their position in the layer stack? Can you also clarify "New timeline"? It kind oof reads that you have the group in an exisiting timeline, and want to use them again in a 'new' timeline.
It looks like you can't. Not only that, but you can't do a work around by linking the layers in the group. Unless you set the stopwatch on all the layers, only the ones that have the stopwatch set will move then the clip is played.