Artboard Alignment

As @Kurt Gold said, you appear to have one item (a group) per artboard.   This simple script should centre these items.  It will ignore artboards with >1 item though. 

var doc = app.activeDocument;
var ABs = doc.artboards;
var output = "";
for (var i = 0; i < ABs.length; i++) {
    ABs.setActiveArtboardIndex(i);
    doc.selectObjectsOnActiveArtboard();
    var d = doc.artboards[i].artboardRect;
    if (doc.selection.length == 1) {
        var w = doc.selection[0].width;
        var h = doc.selection[0].height;
        doc.selection[0].position = [
            (d[0] + ((d[2] - d[0]) / 2)) - w / 2,
            (d[1] + ((d[3] - d[1]) / 2)) + h / 2
        ];
    }
}

Well, Met, this is rather a task for a script. I can imagine that it might also be possible with an action, but it would be pretty convoluted.

One thing that is always a bit arcane is the term "picture". To me it normally means a raster image, but there are many users that use it in terms of any kind of illustrations that can be done in Illustrator. Therefore it's difficult to suggest something without first taking a look at a sample file.