I have run across something that I have not seen before a solid F2 handset.
what I mean by solid, is that there is no hole running through it from the transmitter to the receiver and no wire that I can see going from the transmitter to the receiver.
Are the wires hidden inside the plastic or bakelight ?
Below is a picture of the transmitter side.
All of the earlier F handsets had internal wires molded into the bakelite, instead of wires running through a hollow in the handle. This was true of the F1 handset and the F2 handset and other F handset variations. later models did run separate wires through a hollowed out tunnel, similar to the later G handset.
The F1 handset was made to take a three conductor handset cord that had one transmitter wire, one receiver wire, and a common wire for the other side of both the transmitter and the receiver.
The F2 handset was made to take a 4-conductor handset cord, and was usually used in lieu of a headset at a PBX switchboard where the operator did not need to have a headset at all times, and the handset was more comfortable to use.
The F2 can also be wired to take a 3-conductor handset cord, which is what is used in the 300 series phones like a 302, or your 304. Alternatively, if you have a 4-conductor handset cord connected to the F2 handset, that 4-conductor cord can easily be connected to a 302/304 too.
Here is another link to Paul F's website which describes the variations of the F style handset.
http://www.paul-f.com/we300typ.htm#Fhandsets
You should bookmark this site :)