External monitor has picture interference when laptop is connected to the network?

My external monitor (HDMI) flickers and has picture interference when my Vaio laptop (Windows 10) is connected to the network.

The problem does not occur when the laptop is on battery power.

What to do?

Voltage peaks.

Try it out:

Use a different socket (explicitly on a different current phase) or a lightning protection adapter
Take out the battery and operate directly with electricity only (battery error during charging)
use better shielded HDMI cable
other power supply (this could be a defect)

When things go stupid, it is the laptop's charging electronics that generate the voltage peaks. Here you have the least influence, except to hope for a guarantee.

Thanks. It is always when the CPU usage is just high. The laptop fan also gets very loud. Could that be a sign of the cause?

That is good additional information. Conceivable yes, especially with graphics chips that are linked to the CPU architecture, more likely (e.g. With AMD VEGA).

In pure battery mode, Windows usually does not run in "high-performance mode", so it could also be explained by the fact that your hardware actually can't cope with its maximum load - especially the graphics chip, as it affects the external HDMI source.

Here you could try to suppress the high-performance mode in the energy saving settings of Windows during mains operation. But also try other graphics card and chipset drivers (board) - maybe that will help.

If in doubt, slow down the CPU or GPU gently by 100-300 Mhz using software. Maybe then the flickering will stop?

But definitely get a better shielded HDMI cable. That alone could help.