Is it possible to set up layer on and off in Photoshop action?

It is also possible to record an action to show or hide a layer. However, the issue arises when you cannot select a hidden layer using a keyboard shortcut. This method is preferred for recording steps in an action due to the flexibility it provides when playing the same action across different documents that may have a different layer naming convention but maintain the same layer order. If your documents always have the same layer order and naming convention, then you can record the action by clicking on the layer to select it and then show or hide it. This method records the layer's name as an absolute value, and the action will fail if the document does not have a layer with the exact same name.

"And in case of SAVE AS COPY, it is saved in the action, but there is a problem of overlapping because the name of the file is duplicated."

There are workarounds for this, such as saving in different folders or using Image Processor Pro. The best option, as already mentioned, is Layer Comps. I believe, without verifying, that an action can be recorded to create and save Layer Comps.

And in case of SAVE AS COPY, it is saved in the action, but there is a problem of overlapping because the name of the file is duplicated.

By @jg5235

The File > Automate > Batch command has a checkbox to:

"Override Action "Save As" Commands

https://helpx.adobe.com/photoshop/using/processing-batch-files.html#process_a_batch_of_files

EDIT: I'd just like to make it clear that this override checkbox is for batch processing multiple files, not layers. This is why you need a script to save the visible layers with a variable name, not an action/batch action.

Another option is to create layer comps and use the layer comps to files script.

https://helpx.adobe.com/au/photoshop/using/layer-comps.html

Otherwise, a custom script would be required.

The following scripts can do this:

https://github.com/Paul-Riggott/PS-Scripts/blob/master/Layer%20Saver.jsx

https://github.com/Paul-Riggott/PS-Scripts/blob/master/Layer%20Saver%20Plus.jsx

https://github.com/mechanicious/photoshopCompositionComposer

Info on saving and running scripts:

https://prepression.blogspot.com/2017/11/downloading-and-installing-adobe-scripts.html?m=1