The PATCH method is the correct choice here as you’re updating an existing resource – the group ID. PUT should only be used if you’re replacing a resource in its entirety.
Further information on partial resource modification is available in RFC 5789. Specifically, the PUT method is described as follows:
Several applications extending the Hypertext Transfer Protocol
(HTTP) require a feature to do partial resource modification. The
existing HTTP PUT method only allows a complete replacement of a
document. This proposal adds a new HTTP method, PATCH, to modify an
existing HTTP resource.