LAN cable to WLAN amplifier?

I have a PS4. Of course I have the problem that my download / upload speed with a LAN cable is max 3.1 Mbit / s. The cable (cat7) has to be 20m long to reach the router. My Wi-Fi amplifier would be about 5m from my PS4, what will it do if I attach the LAN cable there?

20 meters shouldn't be a problem for a flat cable. Cat5 cables and better should manage a gigabit / s over 100m. Either your cable is defective or you have other problems.

In industrial plants, LAN cables are significantly longer than 20m. I guess about 100m.

A cable length of 20m is no problem at all, I put a 50m Cat7 here and get 1Gbit / s through cleanly.

I would suspect your problem with the router, it's not the cable and the Wi-Fi doesn't make it any better.

As others already write, 20 meters are no problem at all. I don't own a PS4, but I've heard many times that it has internet speed issues. How is the Playstation connected at the moment?

Connecting to the repeater, if that is possible and it has a corresponding port, is a rather bad idea. The throughput is theoretically and (mostly) practically therefore not much better, if not worse. You can try. Has never hurt. But what would always be advisable is to lay the cable to the router, if possible.