Select layer group below the currently selected group

So you mean that the code should not look for groups inside the currently selected group but outside that? Try the following version

var s2t = stringIDToTypeID;
var t2s = typeIDToStringID;

function getProperty(className, property, index)
{
	var r = new ActionReference(), p = s2t(property)
	r.putProperty(s2t('property'), p)
	if(index != undefined)
		r.putIndex(s2t(className),index)
	else
		r.putEnumerated(s2t(className), s2t('ordinal'), s2t('targetEnum'))
	var a = executeActionGet(r)
	return a
}

function isGroup(layerIdx){
	var r 
	if(layerIdx !== null)
		r = t2s(getProperty("layer", "layerSection", layerIdx).getEnumerationValue(s2t("layerSection")))
	else
		r = t2s(getProperty("layer", "layerSection").getEnumerationValue(s2t("layerSection")))
		
	return r === "layerSectionStart"
}

function movetoEnd(grpIdx){
	var r = t2s(getProperty("layer", "layerSection", grpIdx).getEnumerationValue(s2t("layerSection")))
	while(r !== "layerSectionEnd")
		r = t2s(getProperty("layer", "layerSection", --grpIdx).getEnumerationValue(s2t("layerSection")))
		
	return grpIdx
}

function selectGroup(layerName, layerID) {
    var idselect = stringIDToTypeID("select");
    var desc25 = new ActionDescriptor();
    var idnull = stringIDToTypeID("null");
    var ref11 = new ActionReference();
    var idlayer = stringIDToTypeID("layer");
    ref11.putName(idlayer, layerName);
    desc25.putReference(idnull, ref11);
    var idmakeVisible = stringIDToTypeID("makeVisible");
    desc25.putBoolean(idmakeVisible, false);
    var idlayerID = stringIDToTypeID("layerID");
    var list5 = new ActionList();
    list5.putInteger(layerID);
    desc25.putList(idlayerID, list5);
    executeAction(idselect, desc25, DialogModes.NO);
}

var layerIdx = activeDocument.activeLayer.itemIndex
var bgLayer = getProperty("document", "hasBackgroundLayer").getBoolean(s2t("hasBackgroundLayer")) ? 0 : 1
!bgLayer && layerIdx--
if(isGroup())
{
	layerIdx = movetoEnd(layerIdx)
	r = isGroup(--layerIdx)
	while(!r && layerIdx > bgLayer){
		r = isGroup(--layerIdx)
	}

	if(isGroup(layerIdx)){
		var n = getProperty("layer", "name", layerIdx).getString(s2t("name"))
		selectGroup(n, layerIdx)
	}
}

-Manan