How do I get the summer house to sunflower island?

The Summer House is a 3x3 building which cannot be stored and thus cannot be moved to the outer islands.

You can’t. All buildings that give coins over time are only able to be on your main island called Orchid island.

Any decorations and ground covers can be placed in your ship then accessed at the new island. Only non-coin generating items though.

You can't move the Summer House to Sunflower Island. Any decoration that pays gold coins can not be moved off Orchid Island.