No, there wouldn't be an apostrophe after ITS, the same as you wouldn't put an apostrophe after HIS.
(I put them capitals so as not to confuse with quotation marks, not to shout!!).
The thing is that if the rules were simple there would be an apostrophe his. Or to be fair, since whatever it is belongs to "him", not after "his"; there'd be an apostrophe after "him". Him's wallet.
Isn't there some theory that we used to write e.g. "John his book" and that got abbreviated to "John's book" ? Trouble with that is that if we wrote "Jane her book" it would be "Jane'r book", or is 'her' itself a late development ?