Do I need to register an overseas payroll

My UK resident employer client is about to take on a new employee. He's Greek, working in Greece.

There is a UK PAYE scheme already for other employees.

I told client no tax no NI - unless UK duties off course.

Client wonders about how the overseas employee will pay /  declare tax in Greece. Me stumped.

Just spent 40 mins on phone to HMRC (3 calls) and the most intelligent answers were basically "we don't care about Greek tax" and it's up to the employee to declare?! I countered with surely if it was the other way round - a UK individual employed by non-res employer - then HMRC would expect that employer to set up a UK PAYE scheme. They said no. It's up to that employee to declare.

HMRC basically gave me the basics about NT code and that much!

So does my UK employer client have to tell Greece's tax authorities? My client said they will be employed ( as with no tax/ni it could be easy to make them self employed).

Thank you