[http] Should I use PATCH or PUT in my REST API?

I would recommend using PATCH, because your resource 'group' has many properties but in this case, you are updating only the activation field(partial modification)

according to the RFC5789 (https://tools.ietf.org/html/rfc5789)

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.

Also, in more details,

The difference between the PUT and PATCH requests is reflected in the way the server processes the enclosed entity to modify the resource
identified by the Request-URI. In a PUT request, the enclosed entity is considered to be a modified version of the resource stored on the
origin server, and the client is requesting that the stored version
be replaced. With PATCH, however, the enclosed entity contains a set of instructions describing how a resource currently residing on the
origin server should be modified to produce a new version. The PATCH method affects the resource identified by the Request-URI, and it
also MAY have side effects on other resources; i.e., new resources
may be created, or existing ones modified, by the application of a
PATCH.

PATCH is neither safe nor idempotent as defined by [RFC2616], Section 9.1.

Clients need to choose when to use PATCH rather than PUT. For
example, if the patch document size is larger than the size of the
new resource data that would be used in a PUT, then it might make
sense to use PUT instead of PATCH. A comparison to POST is even more difficult, because POST is used in widely varying ways and can
encompass PUT and PATCH-like operations if the server chooses. If
the operation does not modify the resource identified by the Request- URI in a predictable way, POST should be considered instead of PATCH
or PUT.

The response code for PATCH is

The 204 response code is used because the response does not carry a message body (which a response with the 200 code would have). Note that other success codes could be used as well.

also refer thttp://restcookbook.com/HTTP%20Methods/patch/

Caveat: An API implementing PATCH must patch atomically. It MUST not be possible that resources are half-patched when requested by a GET.

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