What is the simplest solution to use existing unused switch to turn on existing constant outlet?

I have an unused 110v switch that is connected to a ceiling outlet (there is a blank on the ceiling for installing a light or ceiling fan that is not being used).

I have a plug in lamp in the same room and would like a transmitter wired behind the switch or in the ceiling behind the blank because there is plenty of room in the unused junction box in ceiling.

I would then like to wire a receiver to half of the now constantly powered outlet that the lamp is plugged into, toggled by the switch.

What simple transmitter/receiver do you suggest?

I don’t have any home automation in this home and I don’t plan on adding any here. Thank you in advance!

I think the simplest solution is probably going to be not using the wall switch, but instead getting something like an IKEA Tradfi control Outlet kit. These kits come a button and “smart” outlet that are paired to each other and do not require a hub or home automation software (though they can be integrated into Home Assistant and other home automation software).

There are similar kits that include remotes with form factors closer to traditional wall switches, but this forum may not be the best place to ask about those since they often use protocols that don’t work well with automation.

“I don’t have any home automation in this home and I don’t plan on adding any”

Does this mean you are not using Home Assistant and don’t want to do so? If that’s the case, why are you asking this question in the Home Assistant forum?

Or perhaps you mean you do use Home Assistant, but don’t want to mess with creating a complex automation to implement a solution?

Do you have access to the wiring? Like an attic or basement? Because if you have no interest in home automation the best solution would be to just wire the switch to the outlet.