Best way to get a grounded outlet here and keep the light socket?

Wanted to add an outlet adapter to this light socket so I could plug in a couple shop lights and have them controlled by the switch.

Only realized after I bought the adapter that my shop lights have a ground pin, so that won't work.

Im comfortable installing outlets and fixtures, but I haven't dealt with this exact situation before (switch loop to socket, feed through to another outlet down the line) so I'm wondering what's the easiest/smartest way to accomplish what I'm trying to accomplish.

I'm guessing either just make it an outlet and add a socket plug if I want to keep the light, or maybe a lamp holder with a plug and use a splitter for my two shop lights.

Any better ideas?

Also, how should the existing ground be attached? Currently bypasses the socket and goes to the outlet down the line, should it be split to go to the new receptacle and the next outlet separately, or can it run from the new receptacle to the next one?

Thanks.