How to do single sign-on to Sharepoint Server 2010 using credentials from Office 365?

Question

SharePoint

There is a virtual machine with Sharepoint Server 2010 at the initial stage of content filling. There is also a cloud service Office 365, which provides corporate mail. As a corporate mail administrator, I can use Powershell management and download the entire list with accounts.
It is necessary to configure single sign-on in such a way that it would be possible to enter the SP2010 portal using Office 365 accounts. And it is desirable that changes in the list with Office 365 are immediately or a little later synchronized to the Active Directory. I know that you can synchronize the entire list in Office 365, but I didn’t find anything about the reverse / opposite process.