Is it possible? Appears so technically. Natively inside Genesys? I'm thinking no but check with Care, AppFoundry or wait for more responses.
I have not tested the below, but looking at what is available here's an option:
Use the /api/v2/conversations/{conversationId}/participants/{participantId}/digits endpoint. Provided you are not doing a blind transfer, the conversation is still connected to Genesys so you'd be able to send this API call to input DTMF tones to the call. Add a delay for however long so the 3rd party system is ready to receive the tones whenever they are requesting them.
You'd have to subscribe to the v2.analytics.conversation.{id}.metrics notifications API topic to know when a transfer happens and write a trigger that says something like if Topic = Transfer AND DNIS = THIRD_PARTY, run logic to get conversationId / participantId. conversationId is in the schema for this topic, but not participantId. You'd need to make a subsequent call to get that from /api/v2/analytics/conversations/{conversationId}/details.
There are flaws with this like having an agent being required to stay on the phone, which at this point, is equally as good of a solution than doing the above. You'd also have to use the Event Bridge AWS integration since the aforementioned topic cannot be used within Genesys itself.
I'm sure there is more to it than this / someone might have a better solution, but this is just something to get the gears oiled looking at the APIs.
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Caleb Smith
IGS Energy
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