The thing that I'll show you may seem to you ridiculous but that was what I needed.
I wanted somebody to prove me, or explain me that the center of mass would be the center of rotation in the given case.
@Vincent-Lyon.Fr you are absolutely right saying that
"Both forces are vertical, hence by N's second law, no horizontal acceleration of centre of mass."
But that was not that obvious to me until I convinced myself using geometry).
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I got rid of the force of reaction and the remnant forces are shown.
The force F2 that pulls the center of mass to the left is equal in magnitude and opposite to the horizontal component of F1, so there is no any motion there, relative to the horizontal axis, so it gives us the axis of rotation not the point because the center of mass moves relative to the vertical axis of coordinates. To calculate the rotational motion alone we must use P as a central point of rotation because if there were strong enough force of friction the stick would have rotated about it.
Thanks everybody, for the help you gave me, I really appreciate it!