How can I check if a keyboard modifier is pressed (Shift, Ctrl, or Alt)?

It looks like all you need to do is check the keyboardModifiers in your button handler, and select a different action as appropriate. The various modifiers can be OR’d together in order to check for multi-key combinations:

PyQt5:

import sys
from PyQt5 import QtCore, QtWidgets

class Window(QtWidgets.QWidget):
    def __init__(self):
        super().__init__()
        self.button = QtWidgets.QPushButton('Test')
        self.button.clicked.connect(self.handleButton)
        layout = QtWidgets.QVBoxLayout(self)
        layout.addWidget(self.button)

    def handleButton(self):
        modifiers = QtWidgets.QApplication.keyboardModifiers()
        if modifiers == QtCore.Qt.ShiftModifier:
            print('Shift+Click')
        elif modifiers == QtCore.Qt.ControlModifier:
            print('Control+Click')
        elif modifiers == (QtCore.Qt.ControlModifier |
                           QtCore.Qt.ShiftModifier):
            print('Control+Shift+Click')
        else:
            print('Click')

if __name__ == '__main__':

    app = QtWidgets.QApplication(sys.argv)
    window = Window()
    window.show()
    sys.exit(app.exec())

PyQt4:

import sys
from PyQt4 import QtGui, QtCore

class Window(QtGui.QWidget):
    def __init__(self):
        super(Window, self).__init__()
        self.button = QtGui.QPushButton('Test')
        self.button.clicked.connect(self.handleButton)
        layout = QtGui.QVBoxLayout(self)
        layout.addWidget(self.button)

    def handleButton(self):
        modifiers = QtGui.QApplication.keyboardModifiers()
        if modifiers == QtCore.Qt.ShiftModifier:
            print('Shift+Click')
        elif modifiers == QtCore.Qt.ControlModifier:
            print('Control+Click')
        elif modifiers == (QtCore.Qt.ControlModifier |
                           QtCore.Qt.ShiftModifier):
            print('Control+Shift+Click')
        else:
            print('Click')

if __name__ == '__main__':

    app = QtGui.QApplication(sys.argv)
    window = Window()
    window.show()
    sys.exit(app.exec_())