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Custom push-to-talk button on a handheld two-way radio
The typical way these are wired is that the microphone is on the 2.5mm plug - you can see this on the accessories tab on the web site. The ones that are earphone only all have the 3.5mm. Microphone and earphone have 2.5mm and 3.5mm.

The usual way that PTT works is that you haave your PTT-Button in series with your external microphone. When you close the button, the microphone gets connected and pulls down the DC-Voltage that the radio puts on the microphone pin - the radio sees this as PTT and turns on the transmitter.

The typical DC load of an external microphone is around 2.2KOhm. Sometimes higher, sometimes lower, but in that ballpark. I've seen some with 1K, others with 4.7K. If it is too high the radio won't recognize PTT. If it is too low, the microphone signal will be weak and/or distorted.

The wide ring on the plug is ground, the tip is the positive side. Connect the ground wire of the microphone to the wide ring. Connect one side of your PTT button to the tip, the other side of the PTT button to the microphone +. When you push the button, the radio will transmit what comes in through the external microphone. Your external microphone must be an electret microphone - dynamic microphones will NOT work.

The way these work, you cannot use an external PTT with the internal microphone.

ETA: I took a closer look at the accessories page. The 3.5mm plug has three rings instead of two. That is a different system than the one I described. The schematic in the answer below would be the correct description.