Save changes on images coming from Lightroom via script/action

I don't have much experience with script programming in Photoshop yet. So i have a question to the script professionells.

A image was send via “Edit in Photoshop” from Lightroom Classic to Photoshop.

After doing the appropriate changes and editing I like to save and send back the image back to Lightroom with the same name as the original file and in the same folder.. Normally i click on Photoshop’s menu File - Save. I try to do thi with an action or a script.

All attempts so far have failed. Either the user is prompted for a filename, or the photo is saved without such a prompt—but in the wrong directory, as no corresponding query was issued.

My question is how can I simulate the function of the click on File-Save on an image that came from Lightroom.   

I’m on Windows 11 and use Photoshop 27.7 and Lightroom 15.3.1