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In this video we will discuss Why Fugaku didn't let Kakashi join the Coup?
Fugaku was once pressurized by his fellow clansmen to take away Kakashi's Sharingan since he was not an Uchiha.
Though their reasoning made sense, Fugaku just simply dismissed that by saying that they should not make hasty decisions and also should respect Obito’s wishes which were to entrust Kakashi with his Sharingan.
So here is why Fugaku never asked Kakashi to join the coup detat:

Fugaku respected Obito's wishes. This means he understood quite clear that Kakashi was allowed to do with the Sharingan as he saw fit since Obito trusted him with it.
Fugaku is a prideful man. He would never allow an outsider to interfere in his clan's affairs. And he does not like burdening others with his problems too and let alone a stranger.
Fugaku was a good judge of character. This was shown by the fact that he knew all along that Itachi was a pacifist who was against the ways of the Uchiha clan and he had also predicted that they will face off some day due to their different ideology. Fugaku knew pretty well that Kakashi would never agree to doing such an absurd thing as betraying Konohagakure.
Fugaku was not 100% behind the coup detat hence why he send Itachi as a spy to see how the Hogake and the other higher ups were going to do to find a solution for a desirable outcome for both parties. He was hesitant to rebel and he only gave a go ahead due to his demanding clans men’s pressure and as their leader,he knew he had to do something. So yeah Fugaku would have never wanted to involve even more people into that bloodshed.
Fugaku tried by all means to stop chaos being caused by his clansmen. There is nothing that he always tried to avoid more than jeopardizing the Uchiha’s shaky relations with the village. That is why he waited until the last moment to continue with the coup detat plan. So yeah he would not have wanted Kakashi to have been a fuel to the fire of the anslaught by the retaliating village.