How to transfer data from the site example.com to the same one, but only inside the iframe?

Question

JavaScript

Let me explain in more detail:
We have an example.com site, it is built into (for example) google.com via an iframe. At some point, the user clicks on the link (target="_blank") in example.com, which is inside (iframe) google.com, and a new tab for example.com opens.
There, the user performs certain actions and clicks the ready button. We write its actions (data) to localStorage and after clicking done, we give window.close () and return to google.com, where the example.com iframe received data from localStorage and processed it.
The problem is that this scheme does not work in safari, what is the solution?