How to rapidly align one layer to another by center, or simply, align by center?

It would require a script, rather than an action, if you wish to have a single history state.

Select two or more layers, or one layer and a select all then run the following script to align centers with a single history state:

/* 

Align Centers.jsx

www.ps-scripts.com/viewtopic.php?f=66&t=7036&p=35273&hilit=align+layer#p35273

www.macscripter.net/viewtopic.php?id=38890

AdLf = Align Left
AdRg = Align Right
AdCH = Align Centre Horizontal
AdTp = Align Top
AdBt = Align Bottom
AdCV = Align Centre Vertical
*/

#target photoshop

function alignLayers() {

    //Change as required
    alignSelectedLayers('AdCH');
    alignSelectedLayers('AdCV'); 

    function alignSelectedLayers(method) {
        //app.activeDocument.selection.selectAll();
        var desc = new ActionDescriptor();
        var ref = new ActionReference();
        ref.putEnumerated(charIDToTypeID("Lyr "), charIDToTypeID("Ordn"), charIDToTypeID("Trgt"));
        desc.putReference(charIDToTypeID("null"), ref);
        desc.putEnumerated(charIDToTypeID("Usng"), charIDToTypeID("ADSt"), charIDToTypeID(method));
        try {
            executeAction(charIDToTypeID("Algn"), desc, DialogModes.NO);
        } catch (e) {}
        //app.activeDocument.selection.deselect();
    }
}
app.activeDocument.suspendHistory("Align Centers", "alignLayers()");

Or perhaps you  would prefer to have the script play an action set and action so that you have more flexibility while still retaining a single history step:

#target photoshop

function alignLayers() {
var actionName = "My Align Centers Action"; // Action to run
var actionSet = "My Align Set"; // Action set to run
app.doAction(actionName,actionSet);
}
app.activeDocument.suspendHistory("Align Centers", "alignLayers()");

A custom keyboard shortcut can be assigned to an installed script or you could record an installed or browsed script into an action.

Downloading and Installing Adobe Scripts

Thanks.

I created an action that does both Align to Horizontal & Vertical Center. Then I assigned a function hotkey to it.

How to assign a hotkey to action: How do you assign a hotkey to a recorded action?

Then I assigned CTRL SHIFT Q ... and.... CTRL SHIFT 1 .... to align by horizontal center... and the other for vertical center. (Edit--> Keyboard Shortcuts--> )