This could easily be overkill, but since the above answers didn't cover things from a group policy angle here goes. I had already tried "Register PTR Records" to no effect, but "Register DNS records with connection-specific DNS suffix" could actually do the trick, because it indeed does when manually enabling this option in the network connection properties (see the accepted answer). sends option 81 and its fully qualified domain name to the DHCP server and requests the DHCP server to register a pointer resource record (PTR RR) on its behalf.
Without a way for DHCP to interact with DNS, the information maintained by DNS for a DHCP client may be incorrect.
But checking the boxes for "Register this connection's address in DNS" and "Use this connection's DNS suffix in DNS registration" makes it work.
I don't have a convenient test environment to try it...
An additional DHCP option code (option code 81) enables the return of a client's FQDN to the DHCP server.
If implemented, the DHCP server can dynamically update DNS to modify an individual computer's resource records with a DNS server using the dynamic update protocol.