Have you tried pushing the wire thru the brown sleeve to see what is involved? The usual socket treatments are wire is soldered or crimped into a sleeve which inserts in the brown insulator or in non protected sockets, is soldered directly to a metal center button which is part of the phenolic insulator. In insulator acts as the spring seat as well as the base support for the bulb contact.
If wire is soldered, just melt the solder, pull out the wire and clean out the sleeve for the new wire. If crimped you will need to cut the wire close to the sleeve and try and drill out enough of the old wire to leave enough space or a hole you can use to solder the new wire into the old contact.
Here are a couple of examples of typical sockets borrowed from the RI wiring website where you can buy new sockets and some individual pieces if yours are in really bad shape. Yours is probably an older version of socket 1.
Attach file:
socket 1.jpg (54.72 KB)![209_65ff0734eb28c.jpg 772X494 px](https://packardinfo.com/xoops/html/uploads/newbb/209_65ff0734eb28c.jpg)
socket2.jpg (51.75 KB)![209_65ff074575edb.jpg 670X500 px](https://packardinfo.com/xoops/html/uploads/newbb/209_65ff074575edb.jpg)
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