Should one who joined the prayer late repeat the prostration of forgetfulness if he did it with the imam?

If the imam makes a mistake and does the prostration of
forgetfulness before the salaam, the one who joined the prayer late has to
follow him, because he is connected to his imam until he says the salaam.
Then when he makes up whatever he missed, he must prostrate again, because
his prostration with the imam was not done at the right point in the prayer.
The prostration of forgetfulness cannot be done during the prayer, rather
his prostrating with the imam was only because he was following the imam. 

But if the imam’s mistake occurred before the latecomer
joined the prayer, then he should not repeat the prostration, because the
ruling on his imam’s mistake does not apply to him, as it happened before he
joined him in prayer. 

But if the imam does the prostration of forgetfulness after
the salaam, then the latecomer should not prostrate with him, because
following the imam in this case is not possible except by saying the salaam
with him, and the latecomer cannot do that, as he does not say the salaam
until after he has completed his prayer. 

But if the imam’s mistake came before he joined him, then he
does not have to do the prostration of forgetfulness, because the ruling on
his imam’s mistake does not apply to him. But if his mistake came after he
joined him, then he should prostrate after he has said the salaam.  

Some of the argument we have mentioned above is based on
sound reports, such as the idea that the one who is praying behind the imam
should prostrate following his imam, because the Prophet (peace and
blessings of Allaah be upon him) said: “The imam is only appointed to be
followed.” Agreed upon. 

And some of it is based on sound rational thinking, as
explained. See: al-Sharh al-Kabeer ‘ala al-Muqni’ and al-Majmoo’
Sharh al-Muhadhdhab. End quote.