I moved into a house with very thoughtfully placed female HDMI ports behind where the TV is mounted, and then down below on the wall near the floor to route signals through without long cable runs coming from the wall mount.

Given the arrival of the Smart TV, I'd much rather use that HDMI run in the wall as a bridge to bring wired Internet to the TV and avoid quality issues with WiFi (there's no Ethernet port in the wall behind the TV mount as part of the home's built in wired network). Is there a way to adapt a nearby Ethernet port patched to my router to run into the female HDMI port below, have it come out the female HDMI port by the TV, and then convert it again to an RJ45 to plug into the TV's Ethernet port? My understanding is that HDMI cables can carry the Ethernet signal, but I haven't found any adapters for this use case. Plus all my searches turn up info on HDMI over Ethernet (as opposed to the other way around).

Any insight would be a huge help! Thanks.