Unregistering as self-employed (any consequent changes needed)

I will move overseas (probably on a permanent basis). When exactly should I tell HMRC that I want to close my self-employed position: before leaving, on the day I leave, or later on once I am in the overseas country? And if the latter, how soon after leaving? Also, if I open an LTD and keep it dormant (let's call it "XYZ") and I create a website for it, I guess that as soon as I unregister myself as self-employed I will have to change my website footer (currently it is "My name and surname trading as XYZ", not to make my LTD "XYZ" lose its dormant status). Once I move to Italy, I will open a self-employment position or small company (there it is called "p. iva"). However the website will still be accessible from all over the world (including the UK). So, my question is: what should I write in the footer of the English version of the website at this point to avoid confusion? (Notice: I will still keep my LTD although I will live overseas, and it will have to stay dormant, because I need it to protect my IP name). If I write "My Name and Surname trading as XYZ p.iva" would that make the HMRC think I am still working as self-employed in the UK? And what if I keep the footer as it was (i.e. "My Name and Surname trading as XYZ")? Finally, once I close my self-employed position at the HMRC, should I erase any trace of it online? I am talking about Google Maps, social media, and other websites (including mine) where I advertise my previous services as a professional working in the UK. And if I am allowed to keep those profiles, should I change the information to reflect the current circumstances?

Hi Guido,

For the Self-Employment you should notify HMRC once this has ended.
Unfortunatly we can't advise you with regards the rest of your query.

Thank you.