Should I use PATCH or PUT in my REST API?

The R in REST stands for resource

(Which isn't true, because it stands for Representational, but it's a good trick to remember the importance of Resources in REST).

About PUT /groups/api/v1/groups/{group id}/status/activate: you are not updating an "activate". An "activate" is not a thing, it's a verb. Verbs are never good resources. A rule of thumb: if the action, a verb, is in the URL, it probably is not RESTful.

What are you doing instead? Either you are "adding", "removing" or "updating" an activation on a Group, or if you prefer: manipulating a "status"-resource on a Group. Personally, I'd use "activations" because they are less ambiguous than the concept "status": creating a status is ambiguous, creating an activation is not.

• POST /groups/{group id}/activation Creates (or requests the creation of) an activation.
• PATCH /groups/{group id}/activation Updates some details of an existing activation. Since a group has only one activation, we know what activation-resource we are referring to.
• PUT /groups/{group id}/activation Inserts-or-replaces the old activation. Since a group has only one activation, we know what activation-resource we are referring to.
• DELETE /groups/{group id}/activation Will cancel, or remove the activation.

This pattern is useful when the "activation" of a Group has side-effects, such as payments being made, mails being sent and so on. Only POST and PATCH may have such side-effects. When e.g. a deletion of an activation needs to, say, notify users over mail, DELETE is not the right choice; in that case you probably want to create a deactivation resource: POST /groups/{group_id}/deactivation.

It is a good idea to follow these guidelines, because this standard contract makes it very clear for your clients, and all the proxies and layers between the client and you, know when it is safe to retry, and when not. Let's say the client is somewhere with flaky wifi, and its user clicks on "deactivate", which triggers a DELETE: If that fails, the client can simply retry, until it gets a 404, 200 or anything else it can handle. But if it triggers a POST to deactivation it knows not to retry: the POST implies this.
Any client now has a contract, which, when followed, will protect against sending out 42 emails "your group has been deactivated", simply because its HTTP-library kept retrying the call to the backend.

Updating a single attribute: use PATCH

PATCH /groups/{group id}

In case you wish to update an attribute. E.g. the "status" could be an attribute on Groups that can be set. An attribute such as "status" is often a good candidate to limit to a whitelist of values. Examples use some undefined JSON-scheme:

PATCH /groups/{group id} { "attributes": { "status": "active" } }
response: 200 OK

PATCH /groups/{group id} { "attributes": { "status": "deleted" } }
response: 406 Not Acceptable

Replacing the resource, without side-effects use PUT.

PUT /groups/{group id}

In case you wish to replace an entire Group. This does not necessarily mean that the server actually creates a new group and throws the old one out, e.g. the ids might remain the same. But for the clients, this is what PUT can mean: the client should assume he gets an entirely new item, based on the server's response.

The client should, in case of a PUT request, always send the entire resource, having all the data that is needed to create a new item: usually the same data as a POST-create would require.

PUT /groups/{group id} { "attributes": { "status": "active" } }
response: 406 Not Acceptable

PUT /groups/{group id} { "attributes": { "name": .... etc. "status": "active" } }
response: 201 Created or 200 OK, depending on whether we made a new one.

A very important requirement is that PUT is idempotent: if you require side-effects when updating a Group (or changing an activation), you should use PATCH. So, when the update results in e.g. sending out a mail, don't use PUT.

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.