Is there a way to move the group from layer to layer?

For example, Layer 1 has Group 1 and Group 2.
Is there a way to move group 2 to layer 2?

Yes.
The #layer= method is used to set the layer for the drawing element.
Since the Group is a subclass of the drawing element, you can use this method with groups. (Sketchup::Group>Drawingelement>Entity>Object)

Actually you are not moving something to the layer. You are associating entities with different layers in SketchUp

Thank you!