how to move a whole group

Just click on the group in the layers panel, then press ctrl + T , now you can transform the group.

one more tip, with the Move tool , you can right click with the mouse on any object in the canvas, the menu will show the names of the layers and group, you can select the name of the group, then press ctrl + T

I would not use auto select. I would select the group in the layers panel, and I believe you can just select the group and/or the group and all the layers in the group. You need to make sure that all the layers are visible and not licked.