Change your code like -
$('#Global-Button').unbind('click').bind('click', function(event) {
$("#Button-A").trigger("click");
})
and also -
$('#Button-A').unbind('click').bind('click', function() {
$.ajax({
type: 'get',
url: 'some/url/gets/called',
success: function() { this happens on success }
});
});
