Having game sounds played through voice chat

Solution 1:

There is an audio mixing program online that you can use for something like this called VoiceMeeter Banana, available here. This will allow you to add virtual sound inputs/outputs to connect audio sources together, meaning that you can use this to play the game sounds in conjuction with a person speaking on Skype through the same audio source.

My friend and I also use this when we play Keep Talking and Nobody Explodes, and it works perfectly.

Solution 2:

As a sidenote to Texenox's answer, I also suggest you get the Virtual Audio Cable and the Hi-Fi Cable. These will give you more options when it comes to setting up VoiceMeeter Banana.

For example, you can connect one of the Virtual Cables' input to Skype's sound output, feed the same cable's output into one of the hardware inputs, and then run that through the hardware output connected to your headphones.