It is worth noting that Sketchup has two paste options, the standard ctrl v which will paste the selection onto your cursor so you can place it somewhere.
But the other more useful one is Paste In Place which doesn’t attach it to your cursor, rather it simply pastes it exactly where you copied it from. This is very useful when moving things from one model to another, it means they are in exactly the same place without you trying to position them perfectly. It also means you can copy something and paste it into a different context without moving it. So you can create something on the outside of a group and then cut and paste it inside the group.
I don’t think the web version has a default keyboard shortcut for Paste in Place, but it is on the context menu and if you use the little Rabbit search icon to find it it gives you the option to assign a shortcut to it there.