Adding checkable QToolButton to QToolBar with an icon, a text and a menu

Question

Currently, I have a QToolBar with different checkable actions grouped in QActionGroups set to autoExclusive. This works well.

Now, I would like to have one QAction with a menu so that user can choose the behavior of this action. So I decided to use a QToolButton coupled to a QAction

Problem: I found 2 ways to add stuff to QToolBar :

1. adding action: it displays action text(beside icon) without toolbutton submenu
2. adding toolbutton: does not display action text (only icon) but submenu is shown

Below is an MWE:

from PyQt5.QtCore import QSize, Qt
from PyQt5.QtGui import QIcon
from PyQt5.QtWidgets import (QMainWindow, QToolBar, QAction, QApplication, QStyle, 
                             QActionGroup, QToolButton, QMenu)


class MainWindow(QMainWindow):
    def __init__(self):
        super(MainWindow, self).__init__()

        # Toolbar definition
        self.toolBar = QToolBar(self)
        self.toolBar.setIconSize(QSize(24, 24))
        self.toolBar.setToolButtonStyle(Qt.ToolButtonTextBesideIcon)
        self.addToolBar(Qt.TopToolBarArea, self.toolBar)

        # Creation of 2 dummy actions
        a1 = QAction("My action 1", self)
        a2 = QAction("My action 2", self)

        # Making actions checkable
        a1.setCheckable(True)
        a2.setCheckable(True)

        # Creating one ToolButton with one default action
        tB = QToolButton()
        tB.setText("Action 3")
        a3 = QAction("Action 3", self)
        tB.setDefaultAction(a3)
        tB.setCheckable(True)
        a3.setCheckable(True)

        m = QMenu(tB)
        sub_a1 = QAction("sub 1", m)
        sub_a2 = QAction("sub 2", m)
        sub_a1.setCheckable(True)
        sub_a2.setCheckable(True)
        m.addActions([sub_a1, sub_a2])
        tB.setMenu(m)
        tB.setPopupMode(QToolButton.ToolButtonPopupMode.MenuButtonPopup)

        # Adding some icon to each action
        a1.setIcon(QIcon(QApplication.style().standardIcon(QStyle.StandardPixmap.SP_DirIcon)))
        a2.setIcon(QIcon(QApplication.style().standardIcon(QStyle.StandardPixmap.SP_FileIcon)))
        a3.setIcon(QIcon(QApplication.style().standardIcon(QStyle.StandardPixmap.SP_FileIcon)))
        tB.setIcon(QIcon(QApplication.style().standardIcon(QStyle.StandardPixmap.SP_FileIcon)))

        ag = QActionGroup(self)
        ag.setExclusive(True)
        ag.addAction(a1)
        ag.addAction(a2)
        ag.addAction(a3)

        # Two ways to add things to self.toolBar
        self.toolBar.addActions([a1, a2, a3])   # <-- (1) adding a3: displays text "Action 3" (beside icon) without submenu

        self.toolBar.addActions([a1, a2])       # <-- (2) adding tB: does not display text "Action 3" (only icon)
        self.toolBar.addWidget(tB)              #                but submenu is shown


if __name__ == '__main__':
    app = QApplication([])
    mainWindow = MainWindow()
    mainWindow.show()
    app.exec()