There are four wires, two go to the brass screw, one goes to the chrome screw, the last one goes in a hole on the other side of the socket. How do I secure that wire?

When you push the wire in it should stay .
Most of then have a catch inside

What if it doesn't stay?

I tries doubling it back to make it thicker. It just pulls right out. I tried taking a screw from the old one but it doesn't fit. With it in and not all put together, I pluged it in and got a poke touching the top edge of the bulb opening. So now everything is undone and unplugged.