Of course, doing it without telling the new employer is a great way to get an offer rescinded.
Also, if you modify it after the other side signed, then you’ll want them to initial the changes.
If you modified it, then they signed without noticing the modifications, then I guess ask a lawyer.
I don’t see what outsourcing has to do with it. If they have power of attorney to an external HR firm, that seems like their problem. (I’ve never encountered a company dumb enough to do that though.)